Some big generalizations embedded here, but that's kinda what you asked for. There are a bunch of competing effects that mostly balance out. I started trying to write out all the pluses and minuses but as I said, the effects mostly cancel out.
Good design propellers have very flat efficiency curves up to the point where the blade tip speed gets high enough that transonic drag starts to occur.
No reason not to run props up to tip mach no. of 0.75 or so. Beyond that, the trade-off becomes, for a direct drive engine, whether the increased shaft power at higher RPM exceeds the efficiency loss to drive the prop at the higher RPM. GA airplanes pretty routinely get up around M=0.8 tip speed to get max thrust.
Assuming you do not compare to a propeller with nearly sonic tip speeds, then NO, slowing the prop down does not necessarily increase efficiency.
I think we talked about this before. You were trying to select a cam to move the torque curve to lower RPM so you could turn your prop slower and I kept questioning this. If you have an engine wear/reliability concern that says that you want to run the engine slower, and you can't change your gear reduction, then sure, go ahead and design a prop for the RPM that you are comfortable running the engine. You are making the choice to de-rate the power available for reliability reasons. But there is no gain to slowing down any more than that. The efficiency will stay about the same, and the thrust will just go down as the engine power goes down. Or, if you do change the gear reduction so that the engine RPM stays the same and just turn the prop slower, the prop efficiency will stay about the same, the power and thrust will stay about the same.