Existence and realization of the Henselization for almost commutative local rings

Let $A$ be an almost commutative separated local ring. Denote by $\bar{A}$ the intersection of all local Henselian rings $H$ in the completion $\hat{A}$ such that $A\subset H\subset\hat{A}$ and $m_{\hat{A}}\cap H=m_H$. Henselization conjecture. The Henselization $A^h$ exists and

$$A^h\simeq\bar{A}.$$

This extends the commutative characterization of the Henselization as the intersection of the local Henselian rings between $A$ and its completion. The statement concerns existence and identification in the non-commutative almost commutative setting; the supplied source does not indicate whether it has been resolved.