Millson's filtration conjecture for conformal block and invariant rings

Let

be a trivalent tree with $n$ leaves, and let

denote the relevant vector of dominant $sl_2$ weights. Consider the invariant ring

$$\bigoplus V(n\vec{r})^{sl_2(\mathbb{C})}.$$

For the moduli space of quasiparabolic bundles on

$$with marked points$$

, let

$$be the associated line bundle and let$$

be the polytope obtained from the tree polytope by fixing the leaf weights to

$$. **Millson's conjecture.** Give the ring$$

the associated term-order filtration from HMM and HMSV, with associated graded ring isomorphic to

$$\mathbb{C}[P_{\mathcal{T}}(\vec{r})].$$

Then the associated graded ring for the induced filtration on

$$\bigoplus H^0(\mathcal{M}_{\mathbb{P}^1,\vec{p}}(\vec{r},L),\mathcal{L}(n\vec{r},nL))$$

is isomorphic to

$$\mathbb{C}[P_{\mathcal{T}}^*(\vec{r},L)].$$

The conjecture predicts that the term-order degeneration of the invariant ring and the induced degeneration of the corresponding conformal-block ring are governed by the indicated toric semigroup algebras. The supplied text does not state whether this conjecture has been resolved.