The supervisor is responsible for restarting processes, not killing processes. I need more context to give a taegeted answer but my suspicion is that your liveview is crashing for a reason other than the supervisor. You can put logs or breakpoints in your mount function to see when the liveview re-mounts after a crash. You can do the same for all of your event handlers to see which event is causing the LV to crash.

For example: does the LV die on mount, or only upon some interaction?