Higher Specht basis conjecture for the generalized coinvariant ring $R_μ$

Let $n$ be a positive integer, let $\mu\vdash n$, and for each partition $\lambda\vdash n$ let $\mathrm{SSYT}(\lambda,\mu)$ denote the semistandard Young tableaux of shape $\lambda$ and content $\mu$, and let $\mathrm{SYT}(\lambda)$ denote the standard Young tableaux of shape $\lambda$. For tableaux $S$ and $T$ of the same shape, let $F_T^S$ be the associated semistandard higher Specht polynomial, and let $R_\mu$ be the generalized coinvariant ring.

Higher Specht basis conjecture. The set of polynomials

$$\mathcal{B}_\mu=\left\{F_T^S \,:\, (S,T)\in \bigcup_{\lambda\vdash n} \mathrm{SSYT}(\lambda,\mu)\times \mathrm{SYT}(\lambda) \right\}$$

is a basis of $R_\mu$.

This conjecture asserts that the semistandard higher Specht polynomials provide a basis of the generalized coinvariant ring, extending the higher Specht basis phenomenon for related coinvariant constructions. The supplied source does not state whether the conjecture has been resolved.