Canonical Frobenius subalgebra conjecture for flat associative deformations

Let $B$ be a local commutative algebra over $k$ with maximal ideal ${\mathfrak m} \subset B$, and let $\widetilde{A}$ be a flat associative algebra over $B$ such that $A = \widetilde{A}/{\mathfrak m}$ is commutative. Write $B^p$ and $A^p$ for the corresponding Frobenius-power subalgebras. Canonical Frobenius subalgebra conjecture. There exists a canonical subalgebra $\widetilde{A}^p \subset \widetilde{A}$ such that $\widetilde{A}^p$ is flat over $B^p \subset B$, the natural map

$$\widetilde{A}^p \otimes_{B^p} B \to \widetilde{A}$$

is injective, and the quotient map $\widetilde{A} \to A$ induces an isomorphism

$$\widetilde{A}^p/{\mathfrak m}^p \cong (\widetilde{A}^p \otimes_{B^p} B)/{\mathfrak m} \cong A^p \subset A.$$

This conjecture proposes a canonical Frobenius-compatible subalgebra in a flat associative deformation whose special fiber is commutative; it is motivated by the preceding results on Frobenius-constant quantizations, but the supplied text does not indicate whether the conjecture is open or resolved.