Fourfold product stabilization conjecture for Schur functions

Let $\alpha_1\ge\alpha_2\ge\alpha_3\ge\alpha_4\ge 0$ and let $\lambda_1,\lambda_2,\lambda_3,\lambda_4$ be number partitions. For each $i$, write $(n+\alpha_i,\lambda_i)$ for the partition obtained by adjoining $n+\alpha_i$ to $\lambda_i$, and let $s_\mu$ denote the Schur function indexed by a partition $\mu$. The operators $\Delta_m^\lambda$ are the difference operators defined on the corresponding sequences of symmetric functions.

Fourfold product stabilization conjecture.

$$\Delta_3^{(4,4,4)}\Delta_2^{(3,3,2)}\Delta^{(4)}\Delta^{(3,1)}(\Delta^{(2,2)})^2(\Delta^{(2,1,1)})^2\Delta^{(1,1,1,1)}\bigl(s_{(n+\alpha_1,\lambda_1)}s_{(n+\alpha_2,\lambda_2)}s_{(n+\alpha_3,\lambda_3)}s_{(n+\alpha_4,\lambda_4)}\bigr)=0$$

for sufficiently large $n$. Moreover, the multiset $\{\Delta_3^{(4,4,4)},\Delta_2^{(3,3,2)},\Delta^{(4)},\Delta^{(3,1)},\Delta^{(2,2)},\Delta^{(2,2)},\Delta^{(2,1,1)},\Delta^{(2,1,1)},\Delta^{(1,1,1,1)}\}$ is minimal: the sequence is not eventually zero if any one of the difference operators is removed.

The conjecture extends the proved stabilization results for products of two and three Schur functions. The authors explain that their methods support the fourfold case, but a complete proof would require handling a massive number of cases.