Positive Schur-expansion conjecture for triangular partitions

Let $\tau$ be a triangular partition of size $n$, and let $\mathcal{A}_{\tau}(q,t)$ be the associated polynomial. For each partition $\lambda$ of length at most $2$, write $s_{\lambda}(q,t)$ for the corresponding Schur polynomial, and restrict to $|\lambda|\leq |\tau|$. Positive Schur-expansion conjecture. The polynomial $\mathcal{A}_{\tau}(q,t)$ has an expansion

$$\mathcal{A}_{\tau}(q,t)=\sum_{\lambda}c_{\lambda}^{\tau}s_{\lambda}(q,t),$$

where the sum runs over partitions $\lambda$ with length at most $2$ and $|\lambda|\leq |\tau|$, and

$$c_{\lambda}^{\tau}\in\mathbb{N}.$$

This conjecture is supported by extensive calculations for all triangular partitions of size at most $28$, with proofs or representation-theoretic justifications in some instances; the general positive Schur expansion remains open.