Conjecture on infinitely many primes preserving the log canonical threshold

Let $A$ be the coefficient ring in the reduction-modulo-$p$ setup, let $\frak{a}\subset A[x_1,\ldots,x_n]$ be an ideal, let $c$ be its log canonical threshold, and let $c_p$ denote the corresponding $F$-pure threshold after reduction modulo $p$. Infinitely-many-primes conjecture. For every ideal $\frak{a}$ in $A[x_1,\ldots,x_n]$, there are infinitely many primes $p$ for which

$$c_p=c.$$

The preceding results show convergence of $c_p$ to $c$ as $p\to\infty$, while this conjecture asks for equality for infinitely many individual primes. The source does not state a resolution.