Stable affine character conjecture for deformed universal characters

Let $R$ be a dominant sequence of rectangles and let $\lambda$ be a partition. For each of the four stable families indexed by $\diamondsuit\in\{\varnothing,\,\text{cell},\,\text{vertical domino},\,\text{horizontal domino}\}$, let $K^\diamondsuit_{\lambda;R}(t)$ be the corresponding type $A$ deformation coefficient, and let $\overline{M}^{\diamondsuit^t}_{R^t,\lambda^t}(t)$ be the transposed stable fermionic formula. Define $\epsilon=1$ except for $\diamondsuit=\text{cell}$, when $\epsilon=2$. Stable affine character conjecture. One has

$$K^\diamondsuit_{\lambda;R}(t)=\overline{M}^{\diamondsuit^t}_{R^t,\lambda^t}\left(t^{2/\epsilon}\right).$$

At $t=1$ the relationship was essentially known, and the formula is proved for $\diamondsuit=\varnothing$ and for a single rectangle in all nonexceptional affine types; the general dominant-sequence assertion remains conjectural in the supplied text.