Yasuda's smoothness conjecture for higher Nash blowups

Let $k$ be a field of characteristic zero, let $X$ be a variety of dimension $d$, and let $J\subseteq X$ be the Jacobian subscheme. Denote by $J^{(d-1)}$ its $(d-1)$-th neighborhood, namely the closed subscheme defined by the $d$-th power of the Jacobian ideal sheaf. For $[Z]\in \mathbf{Nash}_n(X)$, assume that $Z\nsubseteq J^{(d-1)}$. Yasuda's conjecture. The higher Nash blowup $\mathbf{Nash}_n(X)$ is smooth at $[Z]$. The conjecture concerns desingularization by higher Nash blowups. Together with the fact that, for every closed subscheme $Y\subseteq X$ of dimension less than $d$, all sufficiently large $n$ satisfy $Z\nsubseteq Y$ for every $[Z]\in\mathbf{Nash}_n(X)$, it especially predicts that $\mathbf{Nash}_n(X)$ is smooth for $n\gg0$.