The answer to this is a definite "maybe".

YouTube personality Trent Palmer [tested this very question](https://www.youtube.com/watch?v=24BU15dGdJE). He was actually able to glide *further* with the engine off than he was with it at idle. It was a very small gain, though, on the order of a few seconds extra flying time per 1,000 feet descent. The people he was with reported similar small gains with a stopped propeller. They theorized that a running engine still generates a small P-factor, which has to be compensated for, increasing drag.

It's important to note that all the planes involved in the test were STOL aircraft. Its entirely possible (and indeed *likely*) that other types of aircraft will have different results.