GCD divisibility conjecture for drop functions of the Novelli–Pak–Stoyanovskii algorithm

Let $n\in{\mathbb N}$, let $\lambda$ be a partition of $n$, and let $U\in\operatorname{SYT}(\lambda)$ be such that the corresponding Novelli–Pak–Stoyanovskii algorithm is uniformly distributed. Let $d_U(a,x)$ denote the associated drop function for $1\leq a\leq n$ and $x\in\lambda$. GCD divisibility conjecture. One has

$$\frac{n!}{\operatorname{lcm}\{1,\dots,n\}\cdot\operatorname{gcd}\{d_U(a,x):\;1\leq a\leq n,x\in\lambda\}}\in{\mathbb N}.$$

The conjecture was motivated by computer experiments for row-wise Novelli–Pak–Stoyanovskii algorithms on small partitions, and generalizes the equality established in the paper for the one-row shape. Its status is not resolved in the supplied text.