The central binomial coefficient divisibility conjecture

Let $n$ be an integer greater than $4$, and consider the central binomial coefficient $\binom{2n}{n}$. Divisibility conjecture. The central binomial coefficient $\binom{2n}{n}$ is divisible by $4$ or $9$ for every $n>4$ except $n=64$ and $n=256$. This is the paper's starting problem; the abstract describes the corresponding assertion for $n>256$ as open, and the paper does not provide a definite answer.