The Finitude of Higher Degree Wieferich Prime Numbers?

A prime number $p$ is called a "Wieferich prime number to base $q$" or "$q$-ary Wieferich prime number" granted $$p^2\;\vert\;q^{p-1}-1$$ and hence by a "Wieferich prime number" here it is meant a binary Wieferich prime number. The only known Wieferich prime numbers are $1093$ and $3511$ but the infinitude of the Wieferich prime numbers is widely conjectured. Somewhat conversely, Does there exist $M\in\mathbb{N}$ such that $$p^M\;\vert\;2^{p-1}-1$$ for but finitely many primes $p$ ? Equivalently, is it true that $$\exists(A,B)\in\mathbb{N}\times\mathbb{N}\;\;(M,p)>(A,B)\implies p^M\nmid 2^{p-1}-1$$ which is plainly presented again in asking is it true that $p^M\nmid 2^{p-1}-1$ for all sufficiently large $M\in\mathbb{N}$ and all sufficiently large prime $p$ ? Consider our query with the "binary Wieferich logarithm" or so called "rich log" defined here as $$\mathbf{rig}:\mathbb{N_{0}}\to\mathbb{N_{0}}\;\;\mathbf{rig}:n\mapsto\mathbf{rig}(n)$$ where $\mathbb{N_{0}}:=\mathbb{N}\;\cup\{0\}$ and $\;0^0:=1$ while $\mathbf{rig}(n)$ is understood as the largest $m\in\mathbb{N_{0}}$ such that $$n^m\;\vert\; 2^{n-1}-1$$ Thus, a prime number $p$ is Wieferich if and only if $\mathbf{rig}(p)>1$. It is unknown whether the fiber $$\mathbf{rig}^{-1}[\{1\}]$$ has infinitely many prime members as the infinitude of the non-Wieferich prime numbers to base 2 is an open conjecture where by a "fiber" or "level set" of a function here it is meant that function's inverse image of a singleton set. Is the set of all prime numbers covered with but finitely many fibers of $\mathbf{rig}$ ? Almost any direction on this would be greatly appreciated. Thank you.