Smoothness conjecture for iterated orbit-closure blow-ups

Let $(\mathrm{G},V)$ be a strongly prehomogeneous vector space, meaning an algebraic group $\mathrm{G}$ acts linearly on a finite-dimensional vector space $V$ with finitely many orbits. Suppose its linear orbit diagram consists of orbits $V_{\mathrm{G}}^0,\ldots,V_{\mathrm{G}}^m$, with $V_{\mathrm{G}}^0=\{0\}$ and, after reordering, $V_{\mathrm{G}}^i\subset\overline{V_{\mathrm{G}}^{i+1}}$ for all $i\geq 0$. Let

$$X=\mathbb{P}(\overline{V_{\mathrm{G}}^i})$$

be the projectivization of the closure of any orbit, and consider

$$X_i\stackrel{\pi_i}{\longrightarrow}X_{i-1}\longrightarrow\cdots\longrightarrow X_2\stackrel{\pi_2}{\longrightarrow}X_1=X,$$

where $\pi_k:X_k\rightarrow X_{k-1}$ blows up the strict transform of $\mathbb{P}(\overline{V_{\mathrm{G}}^{k-1}})$ through $\pi_1\circ\cdots\circ\pi_{k-1}$. Smoothness conjecture for iterated orbit-closure blow-ups. The variety $X_i$ is smooth.

This conjecture proposes a uniform smoothness result for resolutions constructed from linear orbit diagrams of strongly prehomogeneous vector spaces. The supplied text states the conjecture but gives no evidence that it has been proved or disproved.