Bostan's conjecture on Catalan numbers modulo 10

Let $C(n)$ denote the $n$th Catalan number. The conjecture concerns which residue classes modulo $10$ are attained by these numbers.

Bostan's conjecture. The following assertions hold:

1. For all $n\geq 0$, $C(n)\mathrel{\not\equiv}3\pmod{10}$.
2. For sufficiently large $n$, $C(n)\mathrel{\not\equiv}1,7,9\pmod{10}$.

This conjecture was observed by Alin Bostan in 2015 and later popularized in 2018. The supplied source presents it as unresolved, in the context of the difficulty of proving analogous residue-avoidance results for non-prime-power moduli.