Affine paving conjecture for punctual noncommutative Hilbert schemes

Let $V$ be the ambient vector space, let $d$ be the number of nodes, and let ${}^0\operatorname{Hilb}^{(m)}(V)$ be the punctual noncommutative Hilbert scheme with resolution $Z^{(m)}(V)$. Let $S_T^{\rm new}$ denote the affine parts indexed by $m$-ary trees $T$ with $d$ nodes. For pairs $\widehat{T}=(T,f)$, where $f:T\rightarrow\{1,\ldots,d\}$ is a compatible ordering satisfying $f(\omega)\leq f(\omega')$ if $\omega\preceq\omega'$, let $S_{\widehat{T}}=S_{(T,f)}$ denote the corresponding affine parts, and let $\pi:Z^{(m)}(V)\rightarrow{}^0\operatorname{Hilb}^{(m)}(V)$ be the resolution map.

Affine paving conjecture. There exist affine pavings of ${}^0\operatorname{Hilb}^{(m)}(V)$ and $Z^{(m)}(V)$ by the parts $S_T^{\rm new}$ and $S_{\widehat{T}}$, respectively, such that $\pi$ maps affine pieces to affine pieces and the fibre over a point in $S_T^{\rm new}$ admits an affine paving by the $S_{(T,f)}$ for $f$ compatible with $T$.

Such pavings would give a geometric stratification compatible with the resolution and describe its fibres through affine pieces indexed by compatible orderings of trees. The supplied text does not state whether the conjecture has been proved or disproved.