Neguț's positivity conjecture for the tableau polynomial $F$

Let $n\geqslant 2$ and let $F(a_2,\ldots,a_n)$ be the polynomial defined by summing the tableau weights over standard tableaux of size $n$.

Neguț's conjecture. If $a_2\geqslant a_3\geqslant \cdots\geqslant a_n\geqslant 0$, then $F(a_2,\ldots,a_n)$ has nonnegative coefficients.

The paper proves this conjecture for $n=2,3$ and $4$ in a slightly more general range, while the assertion for arbitrary $n$ remains open.