Coefficient-Length conjecture for triangular diagonal harmonics

Let $\tau$ be a triangular partition, let $\tau'$ denote its conjugate partition, and let $\boldsymbol{\boldsymbol{\basis}}(\mathcal{E}_{\tau}^{(n)})$ denote the coefficient-length of $\mathcal{E}_{\tau}^{(n)}$. Define

$$\mathcal{A}_{\tau}=\langle \mathcal{E}_{\tau}^{(n)},e_n\rangle.$$

Coefficient-Length conjecture. For all triangular partitions $\tau$,

$$\boldsymbol{\ell}(\mathcal{E}_{\tau}^{(n)})\leq \min(\boldsymbol{\ell}(\tau),\boldsymbol{\ell}(\tau')),$$

and $\mathcal{A}_{\tau}$ does not depend on $n$. This proposes uniform length bounds and stability for the associated symmetric-function specialization; the supplied text does not state whether the conjecture is known or open.