Gleeson (Castle) shot six birdies and three bogeys as he looked set to finish on 68, tied for second, just one shot off leader Perry Cohen, but he struggled on the final hole and came away with a double bogey, which saw him slide down the table.
Hermitage’s Lester enjoyed a less dramatic round, as he cancelled out two birdies with two bogeys, but an eagle on the fifth ensured he got his tournament off to a positive start.
They are both tied for ninth on 70.
Robin Dawson (Tramore) followed up his British Amateur Championship final appearance with a one-under 71. Two birdies on the first two holes ensured a superb start, but his momentum was halted with bogeys on the third and sixth.
However, he recovered well and shot another birdie on the 12th to finish tied for 16th.
The day was a less enjoyable for Naas man Jonathan Yates, who is six shots off the top after a one-over 73.
Portmarnock’s Conor Purcell, who impressed at the Amateur last week, is one shot further back, tied on 59th, while Tiarnan McLarnon (Massereene) is on 76.
Mark Power (Kilkenny) needs a serious upturn in form if he is to challenge after he opened the tournament with a seven-over 79. The winner this week will earn a spot at the 147th Open Championship at Carnoustie next month.
Meanwhile, Jessica Ross finished as the leading Irish qualifier at the Ladies’ British Open Amateur Championship at Hillside, Southport.
The Clandeboye star shot a one-under 71 to add to her opening round tally of 78.
She looked to be going into the turn three-under following three birdies, but a bogey on the ninth dented her score. Another bogey followed on the 12th, but she replied instantly with a terrific birdie on 13.
However, a bogey on the penultimate hole ensured she finished only one under par, though her 36-hole aggregate score of 149 set up a clash with England’s Caitlin Whitehead in match play.
Curtis Cup winner Olivia Mehaffey (Royal County Down) will go up against Jessica Hall after a level-par 72 left her on 150, alongside Annabel Wilson (Lurgan), who hit three bogeys on her way to 76. Wilson’s opponent in the next round will be Yuka Saso of the Philippines.
Kanturk’s Mairead Martin just missed out on a place in the next phase after a late collapse. Heading into her final three holes, she was one-over par, which was enough to take her through, but she hit trouble on 16 and 17, shooting double bogeys on both holes before ending the round with a bogey. She finished two strokes off the pace on 156, 12-over par.
Paula Grant (Lisburn), who led qualifying in last year’s competition, also crashed out after another disappointing round left her a shot behind Martin on 157, while Sinead Sexton (Lahinch) hit 80 to bow out on 158.