# Complexity of comparing extended integer power towers

Inspired by this stackexchange question, is it an open problem to compare two power towers of positive integers if we additionally allow numbers lower in the tower to themselves be represented by power towers? Example:

$$(2^3)^{(4^{5^3})^5}$$

It seems to me like the accepted answer on that page requires a bound on the sizes of the numbers lower in the tower in order to work.