Concavity conjecture for the maximum m-field of the symmetric group

Let $n$ be fixed. For a partition of the symmetric group $\mathbb{S}_n$ into $m$ parts, let $F(n,m)$ denote the maximum attainable sum of intensities, and define

$$f(m)=F(n,m).$$

Concavity conjecture. For any fixed $n$, the function $f(m)=F(n,m)$ is concave.

The conjecture is proposed as a way to understand the growth of $F(n,m)$ beyond the polynomial range, where the paper's upper bounds are far from the available lower bound. No proof is given.