Representation-valued concavity conjecture for stable symmetric-group representations

Let $S_{\infty}=\bigcup_{d\geq0}S_d$, and let $\operatorname{Rep}(S_{\infty})$ be its category of algebraic representations. Its simple objects are $V_{\lambda[\infty]}$, indexed by partitions $\lambda$, and

$$\bar g_{\lambda\mu}^{\nu}=[V_{\lambda[\infty]}\otimes V_{\mu[\infty]}:V_{\nu[\infty]}].$$

Write $X\geq Y$ in the Grothendieck ring $\operatorname{K}(\operatorname{Rep}(S_{\infty}))$ when $X-Y$ is a nonnegative combination of simple objects. Define

$$V:\mathcal P\longrightarrow\operatorname{Rep}(S_{\infty}),\qquad\lambda\longmapsto V_{\lambda[\infty]}.$$

Representation-valued concavity conjecture. The function $V$ is concave under tensor multiplication: whenever $(\lambda+\mu)/2$ is a partition,

$$V_{\frac{\lambda+\mu}{2}[\infty]}^{\otimes2}\geq V_{\lambda[\infty]}\otimes V_{\mu[\infty]}$$

in $\operatorname{K}(\operatorname{Rep}(S_{\infty}))$. This categorifies the preceding coefficient inequality and is intended to explain the disappearance of negative terms after stabilization; its general validity remains open.