The homogeneous-component nonemptiness conjecture for the super-diagonal coinvariant ring

Let $a$, $b$, and $c$ be non-negative integers. Consider the homogeneous component of order $n$ in $\operatorname{SDR}_n$ with $x$-degree $a$, $y$-degree $b$, and $z$-degree $c$. Homogeneous-component nonemptiness conjecture. This component is non-empty if and only if

$$a+b+\binom{c+1}{2}\leq\binom{n}{2}.$$

The claim predicts the precise support of the multigraded Hilbert series of the super-diagonal coinvariant ring. Its status is not resolved in the supplied text.