The non-decreasing stable multiplicities conjecture

Fix a positive integer $k$ and let $\lambda=(a_1,\dots,a_r)$ be a partition of $k$, so that $k=a_1+\dots+a_r$. For each cohomological degree $i$, let $d_i(\lambda)$ denote the stable multiplicity associated with the irreducible $S_k$-representation $V(\lambda)$. Non-decreasing multiplicities conjecture. So long as $k>0$ (equivalently, $V(\lambda)$ is not the family of trivial representations), the sequence

$$d_0(\lambda),d_1(\lambda),\dots$$

is non-decreasing. The preceding results prove vanishing and asymptotic statements for these multiplicities, but the claimed monotonicity is presented as a conjecture based on the computed data.