Lax monoidal product conjecture for derived endomorphism ring spectra

Let a category have a lax or partial lax monoidal product whose structure maps are weak equivalences, and let its unit have a derived endomorphism ring spectrum. Lax monoidal product conjecture. Under suitable technical hypotheses, the lax or partial lax monoidal product induces an $E_{2}$ structure on the derived endomorphism ring spectrum of the unit. This is a proposed converse to the construction of an $E_{2}$ ring spectrum from the smash product on derived categories; the source gives no resolution of the conjecture.