Koszul-duality conjecture for identities of corresponding operations

Let $\mathsf{O}$ and $\mathsf{O}^!$ be a dual pair of binary quadratic non-\Sigma operads. Let $\circ\in\mathsf{O}(2)$ be a bilinear operation and let $\circ'$ be its corresponding operation in $\mathsf{O}^!(2)$. Let $\mathcal{V}$ and $\mathcal{V}'$ be the varieties defined by the multilinear identities of degree at most $3$ satisfied by $\circ$ and $\circ'$, respectively. For each $n$, let $P_n$ and $P'_n$ be the corresponding $S_n$-modules of multilinear polynomials; let $Q_n\subseteq P_n$ and $Q'_n\subseteq P'_n$ be the submodules of identities satisfied by the respective operations, and let $R_n\subseteq Q_n$ and $R'_n\subseteq Q'_n$ be the submodules generated by identities in lower degrees. Koszul-duality conjecture. For all $n\geq 4$, there is an isomorphism of $S_n$-modules

$$Q_n/R_n\cong Q'_n/R'_n.$$

This is a general proposed explanation for the correspondence between new identities in dual operads. The dendriform/pre-Jordan and dialgebra/Jordan-diproduct statement is a special case, while the validity of the general assertion remains open.