The first nonzero stable multiplicity bound conjecture

Fix a positive integer $k$ and let $\lambda=(a_1,\dots,a_r)$ be a partition of $k$, so that $k=a_1+\dots+a_r$. Let $d_0(\lambda),d_1(\lambda),\dots$ be the stable multiplicities associated with $V(\lambda)$, and suppose that $d_l(\lambda)$ is the first nonzero term of this sequence. First nonzero multiplicity conjecture. One has

$$l\leq k.$$

Moreover, equality $l=k$ occurs if and only if

$$\lambda=(1,1,\dots,1),$$

so that $V(\lambda)$ is the family of $k$th exterior powers of the standard representation. This conjecture refines the preceding vanishing results by predicting a sharp bound on the first nonzero degree and characterizing the equality case.