Is there an operad whose algebras are homotopy commutative $E_1$-algebras?

Question

I might guess that the Boardman-Vogt tensor product of the $E_1$ operad and the $A_2$ operad might do the trick. That is, I would guess that an $A_2$ object in $E_1$ algebras, or equivalently an $E_1$ algebra in $H$-spaces, is the same as a homotopy commutative $E_1$ algebra. The idea would be to use the Eckmann-Hilton argument. But I'm not sure.