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In this paper, we investigate the structure of the convergent quantization ofthe 1-shifted cotangent bundle $S$ of a smooth scheme $X$ over a perfect fieldof positive characteristic. The quantization is an $E_2$-algebra over theFrobenius twist $S'$ of the 1-shifted cotangent bundle which restricted to thezero section $X'\rightarrow S'$ is weakly $E_2$-Morita equivalent to thestructure sheaf $\mathscr O_{X'}$ of the Frobenius twist $X'$ of $X$.  Explicitly, we show that the $(\infty,2)$-category of coherent (left-)modulesover $\mathscr O_{X'}$ is equivalent to the full subcategory of the$(\infty,2)$-category of coherent (left-)modules over the quantizationrestricted to the zero section generated by $\mathscr O_X$ are equivalent.

Weak $E_2$-Morita equivalences via quantization of the 1-shifted cotangent bundle