The higher BV quantization conjecture for shifted quadratic modules

Let $X$ be a derived stack and let $n$ be an integer. Write $\mathcal{Q}\operatorname{uad}_{k}(X)$ for the category of $k$-shifted quadratic modules on $X$, and $\mathcal{A}\operatorname{lg}_{\mathrm{E}_{n}}(-)$ for the category of $\mathrm{E}_{n}$-algebras. The Heisenberg Lie algebra is the Lie algebra obtained from a quadratic module by the induced central extension.

Higher BV quantization conjecture. There is a natural equivalence

$$\mathcal{A}\operatorname{lg}_{\mathrm{E}_{n}}(\mathcal{Q}\operatorname{uad}_{1}(X))\simeq \mathcal{Q}\operatorname{uad}_{1-n}(X).$$

Under this equivalence, the induced functor

$$\mathcal{Q}\operatorname{uad}_{1-n}(X)\simeq \mathcal{A}\operatorname{lg}_{\mathrm{E}_{n}}(\mathcal{Q}\operatorname{uad}_{1}(X))\to \mathcal{A}\operatorname{lg}_{\mathrm{E}_{n}}(\mathcal{L}\operatorname{ie}_{1}(X))\simeq \mathcal{L}\operatorname{ie}_{1-n}(X)$$

is a $(1-n)$-shifted version of the Heisenberg Lie algebra, and the composite functor $\mathcal{Q}\operatorname{uad}_{1-n}(X)\to \mathcal{A}\operatorname{lg}_{\mathrm{E}_{n}}(X)$ is the $\mathrm{E}_{n}$-enveloping algebra of the shifted Heisenberg Lie algebra.

This conjecture describes the expected structure of $\mathrm{E}_{n}$-algebras in the category of shifted quadratic modules, extending the analogous description for shifted Lie algebras. The supplied text gives the assertion as an expectation but does not establish it or provide evidence of resolution.