Lam's strengthened square-free positivity conjecture

Let $\lambda$ be a partition, let $\mathfrak{J}_\lambda(\mathbf{u})$ be the associated noncommutative Schur function, and let $I_{\mathrm{L},\le 3}$ and $I_{\mathrm{A},3}$ be the ideals defined in the source. For a reverse semistandard tableau $T\in\operatorname{RSST}(\lambda;N)$, let $\operatorname{sqread}(T)$ denote its square-reading word, and let $\mathbf{u}_{\operatorname{sqread}(T)}$ be the corresponding noncommutative monomial.

Lam's strengthened square-free positivity conjecture. For every partition $\lambda$, $\mathfrak{J}_\lambda(\mathbf{u})$ is $\mathbb{Z}$-monomial positive modulo $I_{\mathrm{L},\le 3}\cap I_{\mathrm{A},3}$, with monomial expansion

$$\mathfrak{J}_\lambda(\mathbf{u})\equiv\sum_{T\in\operatorname{RSST}(\lambda;N)}\mathbf{u}_{\operatorname{sqread}(T)}\pmod{I_{\mathrm{L},\le 3}\cap I_{\mathrm{A},3}}.$$

This is presented as a strengthening of a theorem giving the same square-reading expansion modulo $I_{\mathrm{L},\le 3}$, and as a slight variant of a conjecture in the cited work of Lam and collaborators. The source does not report a resolution.