The additivity conjecture for filtered operads

Let $E_k$ denote the little $k$-disks operad and $BD_k$ the corresponding operad in complete filtered chain complexes. Let $\otimes^{\mathbf{L}}_{BV}$ denote the derived Boardman--Vogt tensor product. For $k\geq 1$, the additivity isomorphism

$$E_{k+1}\simeq E_1\otimes^{\mathbf{L}}_{BV}E_k$$

induces a map of operads

$$BD_{k+1}\simeq E_1\otimes^{\mathbf{L}}_{BV}BD_k.$$

Additivity conjecture. For every $k\geq 1$, the additivity isomorphism $E_{k+1}\simeq E_1\otimes^{\mathbf{L}}_{BV}E_k$ induces the stated map $BD_{k+1}\simeq E_1\otimes^{\mathbf{L}}_{BV}BD_k$ of operads in complete filtered chain complexes. Equivalently, an $A_{\infty}$-algebra in $BD_k$-algebras is naturally a $BD_{k+1}$-algebra. This conjecture would provide the $BD_{k+1}$-algebra structure needed for the quantised shifted relative polyvector construction; the source gives no evidence that it has been resolved.