Multipartition sorting conjecture for stable symmetric-group representations

Let $\lambda^{(1)},\ldots,\lambda^{(n)}$ be partitions, and let $\lambda=\bigcup_i\lambda^{(i)}$ be obtained by rearranging all parts in weakly decreasing order. For a partition $\lambda$, define

$$\lambda^{[i,n]}=(\lambda_i,\lambda_{i+n},\lambda_{i+2n},\ldots).$$

Let $V_{\rho[\infty]}$ be the simple object of $\operatorname{Rep}(S_{\infty})$ indexed by $\rho$, with the coefficientwise Grothendieck-ring order.

Multipartition sorting conjecture.

$$\bigotimes_{i=1}^nV_{\lambda^{[i,n]}[\infty]} \geq \bigotimes_{i=1}^nV_{\lambda^{(i)}[\infty]}.$$

This extends the two-part sorting inequality and a special case of a conjecture of Lascoux, Leclerc, and Thibon. The paper presents it as a consequence one might obtain by repeatedly applying the preceding conjecture; the general statement remains open.