Despite five straight losses in the tournament all is not folorn, one should only cast their memories back to fellow Irish provinces' first few years in at Europe's top table to evaluate fairly Connacht's contribution in their debut season.
In fact both Munster and Leinster experienced big defeats to French opposition in their first few forays into European territory. Added to that, Ulster failed to secure a win in their first season.
And with some single score defeats in December and a strong showing in Toulouse, Connacht captain Gavin Duffy says that the appetite for that first win is through the roof.
"Everyone in the squad is looking forward to the match this Friday. We know the result didn't go our way last weekend even though our performance was much improved on the week before," he said.
"And after two weeks on the road, everyone is itching to get back out in front of our home support and show we're in this competition for a reason."
Head coach Eric Elwood added: "As Munster and Leinster fans have reminded us throughout the course of this journey, the Heineken Cup is a not a forgiving environment and both teams experienced rough rides in their early years.
"Aside from the first 40 minutes against Toulouse at home, every team we've played has known they've been in a battle with us and the margins have been tight."
The Connacht management, players and staff are further calling on the fantastic support throughout the province and country to come out and get behind the team at the Sportground on Friday night.
"Connacht fans have gained the respect of our European counterparts and Friday represents a final opportunity to show all of Europe the passion and rugby knowledge that exists across the province," explained Elwood.
That call is being made to the whole of Ireland to get behind Elwood's men on Friday, where a win for Connacht will guarantee a third quarter-final slot for Ulster ahead of their trip to Clermont Auvergne.
"We have been overwhelmed by the messages of support from across the country over the last few months and we're asking everyone to come down to the Sportsground on Friday and create that special atmosphere that is ever present at rugby venues across Ireland.
"At this stage of the season, we are not afraid to admit that we are up against it in terms of qualification for next season's Heineken Cup.
"That doesn't mean we'll stop trying and our supporters are assured we'll fight to the very end of the Rabodirect PRO12, but as of now this could be our last Heineken Cup match for a while and that gives us huge motivation to go out and get that performance and result.
"We are determined to set the tone on Friday evening with a win and that all four provinces make this a memorable weekend for Irish rugby."
Connacht v Harlequins tickets are available on Ticketmaster.ie and also at the Branch offices on College Road, The tickets are priced from €30.