Nonunital E-infinity structure conjecture for the first derivative spectrum

Let $F \colon \mathcal{J} \to \mathcal{S}$ be lax symmetric monoidal, and let $\Theta^1F$ denote its first derivative spectrum. First derivative multiplication conjecture. The spectrum $\Theta^1 F$ is a nonunital $E_\infty$-ring spectrum.

This is presented as a consequence that would follow from the preceding untwisted multiplication conjecture, using the non-equivariant equivalence between the twisted derivative and an iterated loop of the derivative. The claim concerns the existence of coherent commutative multiplication without a unit on the first derivative spectrum.