Verify email if using username authentication?

I'm building a platform that uses username/password as the means to authentication, but I also require email on signup. The only reason I really need email is for "forgot password" links but it will be used for other communications as well. I'm wondering if I should still require users to verify their emails after signup. (Typically I'd limit features for non-verified but for this particular product it's really all or nothing)

On one hand, it adds some friction to the flow and I could lose customers by forcing them to verify a field that isn't exactly necessary for the functionality of the app.

On the other hand, if they do forget their password and entered a fake email I'm now in a sticky situation of proving their identity to send a reset link to different email, or other issues I haven't thought of that may arise from unverified emails.

So what do you think, verify or not?