Dimension equality conjecture for $W_3$ conformal-block solutions at $c=2$

Let $\mathcal C^{(c=2)}_\varsigma$ be the constructed space spanned by the relevant Specht-polynomial solutions, and let $\mathcal S^{(c=2)}_\varsigma$ be the full solution space of the $W_3$-BPZ equations with parameter $\varsigma$. Dimension equality conjecture.

$$\mathcal C^{(c=2)}_\varsigma = \mathcal S^{(c=2)}_\varsigma.$$

The constructed space is already known to be contained in the full solution space, and its dimension is the Kostka number predicted by the $W_3$ fusion rules. The conjecture asserts that this lower bound is sharp.