The stand-in New Zealand captain has not settled at the crease in his last four innings.
Conway, New Zealand's first innings centurion, grafted to 13 before he edged a ball from Ebadot Hossain on to his pads and it ballooned out to Shadman Islam at gully.
The original call was not out but when overturned on review Ebadot produced his trade mark salute which he says is a tribute to his employers, the Bangladesh Air Force.
Young lived dangerously giving Ebadot hope when he edged the quick just wide of third slip while an edge off Mehidy was dropped by the wicketkeeper.
Bangladesh, sent into bat before lunch on day two, occupied the crease for 176.2 overs before Shoriful Ahmed was the last man out before lunch on Tuesday.
It was an impressive team effort with half-centuries for Mominul Haque, Liton Das, Mahmudul Hasan Joy and Najmul Hossain Shanto as well as a 47 by Mehidy.
When New Zealand thought they had seen off all recognised batsmen, Mehidy and Yasir Ali put on 75 for the seventh wicket.