Infinitude conjecture for admissible Pisano-Legendre indices

Let $a$ and $b$ be integers with $(a,b)\neq(0,0)$, and let $\chi(a,b)$ be the set of integers $k$ for which a $k$-Pisano-Legendre prime relative to $(a,b)$ exists. Infinitude conjecture. For every such pair $(a,b)$,

$$|\chi(a,b)|=\aleph_0.$$

The conjecture asserts that every nonzero initial pair admits infinitely many admissible indices $k$. It is motivated only by the numerical examples described in the source, and its resolution is not supplied.