Lawrence Dallaglio and company, though, are still hanging on, and they can seek inspiration from three of their Guinness Premiership rivals - Leicester, Bath and Saracens - who have all toppled Toulouse in France during previous European Cup campaigns.
Wasps could still sneak through to the last eight by winning their last two games, while hoping Toulouse also come unstuck at Llanelli on Sunday week, but it is a tall order.
"This weekend's game is the biggest challenge you can have as a club, and we want to do ourselves justice following two previous poor performances on the road in the Heineken Cup," said former British & Irish Lions coach McGeechan.
"Saturday is about playing for pride and measuring ourselves against the best in the northern hemisphere."
Since suffering a damaging Pool Six defeat against Llanelli last month, Wasps have reeled off four straight wins in domestic and European competition, averaging 35 points a game.
"It is important for us to keep that momentum going," said Wasps' England centre Stuart Abbott, who continues his midfield partnership with Josh Lewsey on Saturday.
"Toulouse are a team full of very good players, though, and it will take a monumental effort on our part, both physically and mentally, to beat them on their own turf."
McGeechan has recalled half-backs Matt Dawson and Alex King, with Joe Worsley reverting to blindside flanker as Jonny O'Connor returns in the number seven jersey.
Leicester's quarter-final ambitions are far more straightforward, with a minimum eight-point victory over Welford Road visitors Stade Francais on Sunday likely to be enough for top spot in Pool Three.
Stade, though, ended Tigers' European Cup ambitions two seasons ago when they won 26-13 in the east midlands, and a repeat performance cannot be discounted.
"The great thing for us at the moment is that everything is in our own hands, and we will only have ourselves to blame if it goes wrong," said Tigers head coach Pat Howard.
Jonah Lomu and his Cardiff Blues team-mates will move to within sight of a first European Cup quarter-final appearance since 2001 if they beat Arms Park visitors Perpignan.
Cardiff have won as many games in this season's tournament than during the last three European campaigns, while Pool Two rivals Perpignan can report just a solitary European Cup victory on Welsh soil - against Neath in 1998.
The force, then, appears to be with Lomu and Cardiff, guaranteeing another bumper five-figure crowd for the New Zealand superstar's latest Blues appearance.
"Jonah has an aura around him," said Scott Robertson, the Perpignan number eight and Lomu's former All Blacks colleague.
"It is true he has lost some of what he did before, and he is still coming back, but he has something special about him.
"Regardless of his form, he is a rugby legend, and legends have an ability to bring out something in a team, both their own and the opposition.
"In terms of changing our preparation to deal with Jonah, although a team should never focus on a single player, Jonah is pretty exceptional.
"He has a great ability to break tackles and draw defenders, and that is something you can't ignore."
Cardiff could conceivably wrap up the group on Saturday, but they would need to win with a bonus point, deny Perpignan a losing bonus and hope Leeds do not collect a five-point maximum from their clash against Calvisano in Italy.
The Tykes, fresh from appointing former Leeds Rhinos boss Daryl Powell as their new first team coach this week, should encounter few problems at Centro Sportivo San Michele, a ground where Cardiff and Perpignan have already won this season.
Elsewhere on Saturday, Leinster must beat Pool Five visitors Glasgow to keep alive their quarter-final hopes, but fading Welsh challengers Llanelli and the Ospreys have little at stake from respective battles with Edinburgh and Clermont Auvergne.
Sunday's action, meanwhile, should see Premiership leaders Sale Sharks close in on a quarter-final berth by beating Pool One strugglers Newport Gwent Dragons at Edgeley Park, while Pool Four contenders Saracens will target five points against Vicarage Road opponents Treviso.