Quadratic lower-bound conjecture for k-Pisano-Legendre primes

Let $a$ and $b$ be integers with $(a,b)\neq(0,0)$. Let $\chi(a,b)$ denote the set of integers $k$ for which a $k$-Pisano-Legendre prime relative to $(a,b)$ exists, and define

$$\zeta_{(a,b)}(k)=\min\{p:p\text{ is a }k\text{-Pisano-Legendre prime relative to }(a,b)\}.$$

Quadratic lower-bound conjecture. For every such $a$ and $b$, there exists a real number $c$ with $0<c<1$ such that

$$\zeta_{(a,b)}(k)>ck^2$$

for all $k\in\chi(a,b)$. The numerical evidence in the paper suggests quadratic growth of the least relevant odd prime, but no proof or resolution is given.