They travel more in hope than expectation when it comes to their aspirations of squeezing into the quarter-finals as one of three best runners-up, with essentially a bonus point win the requirement as well as other results over the weekend going their way.
Yes, it's been that sort of season for Leicester Tigers with injuries - Manu Tuilagi's travails have been well noted - and dips in form having afflicted Richard Cockerill's side for some time with the current push to battle their way into the Premiership play-offs, where they are sixth and one point adrift, now essentially being their main focus.
Sounds not unlike the way things have panned out at the Kingspan of late though at least, in European terms, Leicester managed to just about see off Toulon at Welford Road and then not ship a container-load of points in the return game which, after beating the Scarlets with maximum points, has seen them arrive at the final weekend of pool action with a chance of continuing their interest in the competition.
But there have been some rather unwanted distractions this week which may partially dilute the intended mindset for tonight. First off, full-back Mathew Tait announced that he would be shipping out to hook up with Bayonne for next season, leaving the ever quotable Cockerill rather lost for words.
"We wish him well and there is nothing else we can do about that," the coach uttered while mentioning in a typical aside that maybe he should put his other players on short-term notice deals as Tait has been playing with considerable flair of late.
Strangely, though, the coach had no comment to make on the shock announcement that flanker Jamie Gibson will be joining bitter rivals Northampton Saints at the end of the season which has clearly had quite an impact, particularly with fellow back-rowers Tom Croft and Julian Salvi also out of contract at season's end.
Needless to say, Cockerill has also been voluble on the task ahead this evening when the Tigers go in search of their first European win in Belfast and with a bonus point to boot as they must aim to get all that they can from this final pool game if they are to ultimately stay alive in the competition.
"We will have to throw the kitchen sink at Ulster and chance our arm," he made known while touching on the fact that making the last eight is still a long shot and that simply winning without the accompanying bonus point is not likely to be sufficient.
With the notion that they have to go for the max, the Tigers are fielding a strong side tonight with the only changes from their 40-23 bonus point win over the Scarlets seeing prop Marcos Ayerza and England and Lions hooker Tom Youngs brought in after being benched against the Welsh side.
Second row Graham Kitchener continues in the injury enforced absence of Brad Thorn in a side which includes some big hitters up front in Geoff Parling, Salvi and, of course, Croft.
Behind the scrum, the Tigers will be hoping to unleash the power of Vereniki Goneva while Tait's incursions in the line or indeed in broken play can also bring considerable cutting edge, as will the sniping of Ben Youngs from scrum-half.
The Tigers will certainly unleash everything they can in the hope that they will have done enough to sneak through.