“Holding faith, and a good conscience; which some having put away concerning faith have made shipwreck:” (I Timothy I:17-20)
The problem here has to do with an absence of faith rather than conscience.
And it is a common problem that sincere believers must face. Some more than others.
For a good portion of my young Christian life, I was afflicted by a kind of hyper conscience over a great many things. And on a couple of occasions, it almost destroyed me.