Sorting conjecture for stable symmetric-group representations

Let $\lambda$ and $\mu$ be partitions. Rearrange all their parts in weakly decreasing order as

$$\lambda\cup\mu=(\nu_1,\nu_2,\nu_3,\ldots).$$

Define $\operatorname{sort}_1(\lambda,\mu)=(\nu_1,\nu_3,\nu_5,\ldots)$ and $\operatorname{sort}_2(\lambda,\mu)=(\nu_2,\nu_4,\nu_6,\ldots)$. In $\operatorname{Rep}(S_{\infty})$, let $V_{\lambda[\infty]}$ denote the simple object indexed by $\lambda$, and order the Grothendieck ring coefficientwise.

Sorting conjecture.

$$V_{\operatorname{sort}_1(\lambda,\mu)[\infty]}\otimes V_{\operatorname{sort}_2(\lambda,\mu)[\infty]} \geq V_{\lambda[\infty]}\otimes V_{\mu[\infty]}.$$

This generalizes the cited sorting conjecture for tensor-product multiplicities. Its validity would give a broad family of coefficientwise inequalities in the stable representation category; the paper does not establish it in general.