Bergeron–Favreau–Krob conjecture on differential equations for bounded-height tableau generating functions

Let $Y_k(t)$ denote the generating function for standard Young tableaux of height at most $k$, and let $p_m(t)$ be polynomials in $t$.

Bergeron–Favreau–Krob conjecture. For each $k$, there are polynomials $p_m(t)$ of degree at most $\lfloor\frac{k}{2}\rfloor$ such that $Y_k(t)$ is a solution of a linear differential equation of order at most $\lfloor\frac{k}{2}\rfloor+1$ with coefficients $p_m(t)$.

This conjecture concerns the differential equations satisfied by the D-finite generating functions $Y_k(t)$. It was formulated from computer experiments and analysis of determinant expressions involving modified Bessel functions; the supplied source does not state whether it has been resolved.