Combinatorial Foulkes conjecture for transpose-symmetric matrices

Let $M(n,m)$ be the set of $n\times m$ nonnegative integer matrices with every row and column sum equal to $m$ and $n$, respectively, as in the paper, and let $T(n,m)=\{A\in M(n,m)\mid A\sim A^{\mathsf T}\}$, where $A\sim B$ means that $A$ can be transformed into $B$ by row and column permutations. Combinatorial Foulkes conjecture. If $n\leq m$, then

$$\# T(n,m)/{\sim}\leq\# T(m,n)/{\sim}.$$

By the preceding combinatorial interpretation, this is equivalent to Foulkes' inequality for the sums of plethysm multiplicities. The supplied text does not establish a resolution of this formulation; it is a restatement of the preceding conjecture rather than a separate conjecture.