The blow-up conjecture for the rank-three logarithmic-connection moduli space

Let $\Sigma$ be the Riemann sphere with $n$ parabolic points, fix generic eigenvalues $\left\{\rho_{i,k}\right\}_{i=0,k=1}^{n,3}$, and let ${\mathcal M}$ be the moduli space of logarithmic connections with these eigenvalues. Let $V$ be the $\mathop{\mathrm{Sym}}(N)$-invariant affine subvariety appearing in the preceding theorem, let $\Delta$ be the discriminant locus, and let $\mathop{\mathrm{Sym}}(N)$ act by permuting the $N$ factors. The blow-up conjecture. There exists a nonempty $\mathop{\mathrm{Sym}}(N)$-invariant affine subvariety $\hat V\subset T^*(\mathbb C\setminus P)^N$ which intersects $V$ generically transversely, together with an open subset $(V\cap\hat V)^0\subset V\cap\hat V$, such that

$${\mathcal M}^1:=\operatorname{Blow}_{(V\cap\hat V)^0}\left(\frac{T^*(\mathbb C\setminus P)^N\setminus\Delta}{\mathop{\mathrm{Sym}}(N)}\right)\setminus\widetilde V\subseteq {\mathcal M},$$

where $\widetilde V$ is the proper transform of $V$; moreover, ${\mathcal M}^0\subseteq{\mathcal M}^1$. The conjecture proposes a birational construction of a nontrivial part of the moduli space by resolving the discriminant locus; the paper verifies it in a numerical example for $n=3$, while the general assertion remains open.