Yasuda's local smoothness conjecture for higher Nash blowups

Let $R$ be a local complete Noetherian domain with coefficient field $k$, and set $X:=\operatorname{Spec}R$. For every $n\in\mathbb{N}_0$, consider the $n$-th Nash blowup $\mathbf{Nash}_n(X)$, which is asserted to be well-defined even when $X$ is not algebraizable. Let $J_X$ be the Jacobian subscheme and $J_X^{(d-1)}$ its $(d-1)$-th neighborhood. Yasuda's local conjecture. If $[Z]\in\mathbf{Nash}_n(X)$ satisfies $Z\nsubseteq J_X^{(d-1)}$, then $\mathbf{Nash}_n(X)$ is regular at $[Z]$. The statement is presented as a reduction of Yasuda's global conjecture to the local complete domain case, and also includes the well-definedness assertion for higher Nash blowups of non-algebraizable formal schemes.