Now that Leinster's hot streak has bitten the Dublin dust, Ireland's northern province are out on their own, the only team among the 24 starters to have negotiated the first four rounds with four wins.
Even if they keep winning, against Montpellier at Ravenhill next month followed by the potential pool decider against Leicester in Tigerville one week later, Ulster will still be swimming against the tide of history.
Over the last 16 seasons, 11 teams have stormed into the knock-out stages with a six-out-of-six sweep through the pool competition. And every single one has failed to go the distance.
Only two went as far as the final - Clermont last season when they came up short against Toulon in Dublin and Northampton in 2011 only for the Saints to be engulfed by Leinster's second-half recovery in Cardiff where Irish teams have won the last three finals staged in the Welsh capital.
Three more of those who sailed through their pool matches fell in the semi-finals, none more excruciatingly so than Cardiff Blues in 2009.
As due reward for eliminating the holders, Toulouse, in the last eight, they moved a matter of metres from the Arms Park to the Millennium Stadium for a home semi against Leicester.
Instead of a safe passage into the final, the rugby gods condemned them to the ultimate in close shaves.
Extra-time having failed to separate the teams, the first penalty shoot-out, and surely the last, took place, a gut-wrenching ordeal ultimately settled by the 16th goalkicker, Leicester's Jordan Crane.
The Scarlets had won seven in a row when they came to grief at Leicester in the semi in 2007, a fate which had befallen Leinster against Perpignan at Lansdowne Road two years earlier. The other six all-conquering pool winners failed to get past the quarter-finals.
Ulster, on track to go into the knock-out stages as the No.1 qualifiers, know only too well how one setback can derail a whole campaign.
One defeat last season, to Northampton in Belfast, condemned them to a quarter-final on the road, at Twickenham where Saracens turned the journey into a dead-end.
Even if, as expected, Johann Muller's team make it five out of five against Montpellier at Ravenhill in the New Year, Ulster's cherished wish to return there for the knock-out stage will almost certainly hinge on getting something from the potential pool decider at Leicester.
Ulster supporters do not need reminding that an extraordinary home win over the English Tigers can always be trumped by an even more extraordinary away defeat.
In the space of a week ten seasons ago, Ulster trounced Leicester 33-0 at Ravenhill only to be routed 49-7 in the return at Welford Road, a swing of 75 points.
Despite Leinster falling victim to a 42-point swing at the hands and feet of Northampton, the RaboDirect PRO12 champions are one of three RaboDirect PRO12 teams still on course for the last eight.
Despite some heroics by the Cardiff Blues at Glasgow followed by Edinburgh at Gloucester 24 hours later, the trio heading towards the last eight are all Irish.
The most climactic of finales at Perpignan served only to show that Munster's breathtaking capacity for turning a lost cause into a startling victory knows no bounds.
Just when all looked hopeless after two misplaced passes, Tommy O'Donnell found room where hardly any existed to send JJ Hanrahan off for the winner.
And just to remind everyone that there's no business like European Cup business for the RaboDirect PRO12 clubs, Rhys Patchell kicked a goal for Cardiff Blues against the Warriors at Scotstoun from 62 metres. It sailed so high over the bar that it almost cleared the dead-ball line, a total of 72 metres.
That would make it longer than Paul Thorburn's phenomenal penalty for Wales against Scotland at Cardiff Arms Park in the mid-Eighties, longer than any measured goal since the Springbok full-back Gerry Brand dropped one against England at Twickenham in 1932.
That, would you believe, reportedly travelled 85 yards before coming down somewhere in the crowd.
How they could be sure it went that far, heaven only knows…
The eight qualifiers, as they stand after four rounds:
1 - Ulster (18 pts, 13 tries)
2 - Toulouse (15 pts, 15 tries)
3 - Clermont (15 pts, 11 tries)
4 - Toulon (15 pts, 11 tries)
5 - Munster (14 pts, 11 tries)
6 - Leinster (13 pts, 8 tries)
7 - Saracens (15 pts, 17 tries)
8 - Leicester (15 pts, 11 tries)
(Toulouse ranked above Saracens by virtue of having beaten them at Wembley. Clermont ranked above Toulon on points-difference.)
Teams winning all six pool matches and what happened to them:
2012-3
Clermont - runners-up, beaten by Toulon in Dublin final.
Harlequins - quarter-finalists, knocked out by Munster at The Stoop.
2011-2
Munster - quarter-finalists, knocked out by Ulster at Thomond Park.
2010-11
Northampton Saints - runners-up, beaten by Leinster in final at Millennium Stadium.
2008-9
Cardiff Blues - semi-finalists, knocked out by Leicester in penalty shoot-out at Millennium Stadium.
2006-7
Biarritz Olympique - quarter-finalists, knocked out by Northampton at San Sebastian.
Llanelli Scarlets - semi-finalists, beaten by Leicester at Leicester City FC.
2004-5
Leinster - quarter-finalists, knocked out by Leicester at Lansdowne Road.
2002-3
Leinster - semi-finalists, beaten by Perpignan at Lansdowne Road.
2001-2
Bath - quarter-finalists, knocked out by Llanelli at The Rec.
1997-8
Wasps - quarter-finalists, knocked out by Brive at Loftus Road.
Team of the weekend:
15 Jared Payne (Ulster)
14 Maxime Medard (Toulouse)
13 George North (Northampton)
12 Luther Burrell (Northampton)
11 Napolioni Nalaga (Clermont)
10 Matt Giteau (Toulon)
9 Ben Youngs (Leicester)
1 Thomas Domingo (Clermont)
2 Dylan Hartley (Northampton)
3 Census Johnston (Toulouse)
4 Dan Tuohy (Ulster)
5 Courtney Lawes (Northampton)
6 Calum Clark (Northampton)
7 Tommy O'Donnell (Munster)
8 Samu Manoa (Northampton)
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