Here $a,b,c,d,e$ are distinct and all greater than $1$.

This question was formerly posted on Math.Stackexchange, precisely here, but seems to be more general than some other tough number theory problems found there.

Modular arithmetic does not seem helpful and I haven't seen many theories relating to exponential diophantine equations where the base(s) are variables as well or where the variables are "permuted" and taken as sums. These considerations give rise to the following related subquestions:

Are there any equations that are simpler and similar to this one where more information is known?

What if $e$ is fixed, or perhaps specifically $e=2$?

The motivation is that there are clearly very many solutions (many simple) even when bounding the solutions by a small constant when one allows lack of distinctness, but seemingly none with distinctness.