Are there infinitely many positive integers which are neither a sum nor a difference of two perfect powers?

This question was proposed some years ago at KoMaL.

It's easy to see that the odd numbers can be written as the difference of two consecutive perfect squares.

A similiar problem was posted at MSE and the same problem was posted at MSE. According to OEIS all numbers greater than $426$ and bellow $10^{19}$ can be written as the difference of two powers.

I'm looking for a proof or any reference of this result.
Any help would be appreciated, thanks.