Schur positivity of equal-factor products of chains

Determine whether the product of two equal chains, namely the distributive lattice n\times n, is Schur positive for every n\ge 8.

Background

The paper studies Schur positivity of incomparability graphs of products of two finite chains, denoted m\times n. Its main results establish non-Schur-positivity in the parameter range relevant to the reproduced conjecture of Li, Qiu, Yang, and Zhang, under a two-case interpretation that includes n=2, m\ge8 and n\ge3, m\ge n+5.

The authors explicitly distinguish that result from another prediction, cited as Conjecture 5.2 of Li, Qiu, Yang, and Zhang, concerning equal-factor products. The cited prediction is that products with m=n and n\ge8 are Schur positive; the present paper does not resolve that question. Because the source uses the chain notation \mathbf n\times\mathbf n, the problem asks for Schur positivity of the incomparability graph associated with that product.