Blumberg–Hill conjecture on tensor products of equivariant commutative operads

Let $G$ be a finite group, and let $I$ and $J$ be indexing categories. Write $\cN_{I\infty}^{\otimes}$ and $\cN_{J\infty}^{\otimes}$ for the corresponding equivariant commutative $G$-operads, and let $\obv$ denote the Boardman–Vogt tensor product. Blumberg–Hill conjecture. There is an equivalence

$$\cN_{I\infty}^{\otimes} \obv \cN_{J\infty}^{\otimes} \simeq \cN_{I \vee J \infty}^{\otimes}.$$

The tensor product corepresents pairs of homotopy-coherently interchanging $I$- and $J$-commutative algebra structures, while $I\vee J$ combines the corresponding indexing data. The paper states that its main theorem confirms the conjecture in $\Op_G$, so the conjecture is solved in the setting addressed by the paper.