Rank conjecture for X-base Fibonacci-Wieferich primes

Let $X=\{a_1,\ldots,a_m\}$ be a finite set of algebraic numbers, and let $K$ be a number field containing all $a_i$. An $X$-base Fibonacci-Wieferich prime is a prime ideal $\mathfrak p$ of $K$ such that

$$a_i^{N_{K/\mathbb Q}(\mathfrak p)-1}\equiv 1\pmod{\mathfrak p^2}\qquad\text{for every }i.$$

Let $\langle a_1,\ldots,a_m\rangle$ denote the multiplicative group generated by the elements of $X$, and let its free rank mean the rank of its free abelian part. Rank conjecture for $X$-base Fibonacci-Wieferich primes. If the free rank of $\langle a_1,\ldots,a_m\rangle$ is $1$, then there are infinitely many $X$-base Fibonacci-Wieferich primes; if that free rank is greater than $1$, then there are finitely many.

This heuristic conjecture is presented as contradicting Wall's conjecture in the Fibonacci specialization. The supplied text gives no resolution of the conjecture.