LLT-type Schur expansion conjecture for the stable Kronecker quotient

Let $\mathcal{U}_\varnothing$ be the subalgebra generated by the uncolored variables, let $\mathfrak{J}^\varnothing_\nu$ be the corresponding noncommutative Schur function for a partition $\nu$, let $\mathrm{SYT}'_\nu$ be the specified set of standard tableaux, and let $\operatorname{sqreadLLT}(T)$ be the associated reading word. Let $I_{\mathrm{KR},\leq 3}^{\mathrm{st}}$ be the two-sided ideal of $\mathcal{U}_\varnothing$ generated by the displayed relations with parameter $k=3$. Stable Kronecker expansion conjecture. For every partition $\nu$,

$$\mathfrak{J}^\varnothing_\nu=\sum_{T\in\mathrm{SYT}'_\nu}\operatorname{sqreadLLT}(T)\quad\text{in }\mathcal{U}_\varnothing/I_{\mathrm{KR},\leq 3}^{\mathrm{st}}.$$

The paper presents this as a conjecture related to the Kron-Knuth conjecture and to results of Blasiak, Lam, and collaborators; no resolution is supplied.