[2][4] Conway received a Juris Doctor from Marquette University Law School in 2002.

[3] From 2002 to 2006, Conway worked as an associate at the Milwaukee law firm Gimbel Reilly Guerin & Brown LLP.

[3] In June 2023, Conway and Circuit Judge Marc Aaron Hammer were recommended to the Biden administration by Senators Baldwin and Johnson to fill the vacancy on the U.S. District Court for Eastern Wisconsin.

President Biden nominated Conway to the seat vacated by Judge William C. Griesbach, who assumed senior status on December 31, 2019.

[8][9] On September 25, 2024, the United States Senate invoked cloture on his nomination by a 58–37 vote.