The UV-finite theory conjecture on effective quantum master equations

In the BV formalism, let $I$ be a local interaction for a UV-finite theory, and let $r\to 0$ denote the removal of the regularization. Write $l_n^\hbar$ for the multilinear operations obtained from the renormalization procedure, and let $\hbar$ be the formal quantum parameter. UV-finite theory conjecture. The effective quantum master equation in the limit $r\to 0$ should be expressible as

$$l_1^\hbar I+\frac{1}{2}l_2^\hbar(I,I)+\frac{1}{3!}l_3^\hbar(I,I,I)+\cdots=0,$$

where $\{l_1^\hbar,l_2^\hbar,\ldots\}$ defines a family of $L_\infty$-algebras parametrized by $\hbar$, with these operations replacing the BV Laplacian $\Delta$ through the renormalization procedure. This is presented as an expected structural description of the effective QME for UV-finite theories; the supplied passage does not state whether the claim has been proved or identify any unresolved cases.