Untwisted multiplication conjecture for derivative spectra

Let $F \colon \mathcal{J} \to \mathcal{S}_*$ be lax symmetric monoidal, and let $D_{O(n)}$ denote the dualizing spectrum of $O(n)$. Define

$$\widetilde{\Theta^n}F:= \Theta^nF \otimes D_{O(n)}^\vee.$$

Untwisted multiplication conjecture. The maps constructed from the monoidal structure make $\widetilde{\Theta^n}F$ into a nonunital $E_\infty$-ring spectrum.

This conjecture asserts that twisting the derivative spectrum by the inverse dualizing spectrum removes the equivariant twist from the multiplication. The preceding result supplies the underlying multiplication maps and the source subsequently notes that the conjecture would imply the corresponding statement that $\Theta^1F$ is a nonunital $E_\infty$-ring spectrum.