Recall that an $E_{n,m}$ algebra is an $A_m$ algebra in $E_n$ algebras. Here I index my $A_m$ algebras so that an $A_1$ algebra is pointed, an $A_2$ algebra has a unital multiplication, $A_3$ is homotopy associative, etc. In particular, an $E_n$ algebra is on the one hand an $E_{n,1}$ algebra and on the other hand an $E_{n-1,\infty}$ algebra.

Question: Suppose $X$ is a homotopy $k$-type and an $E_{n,m}$-algebra, when does it have a canonical $E_{n+1}$ structure? In other words, how connected is the map of operads from the $E_{n+1}$-operad to the $E_{n,m}$-operad? What if $m=2$?

Let me also explain briefly where this question comes from and why I'm particularly interested in the $m=2$ case.

A well-known theorem about monoidal categories says:

Theorem: A monoidal category $\mathcal{C}$ is braided iff there's a monoidal splitting of the canonical forgetful map from the Drinfeld center $Z(\mathcal{C}) \rightarrow \mathcal{C}$.

I was wondering what the appropriate generalization of this statement is to general $E_n$ algebras. That is for which $k$-types do $E_{n+1}$ structures on $E_n$ algebras correspond to homotopy splittings of the forgetful map from the $E_{n+1}$ center $Z(A) \rightarrow A$. If you think through the $E_0$ case, it's not difficult to see that a homotopy retract of an $E_1$ algebra is only an $A_2$ algebra. Applying this observation to $E_0$ algebras in $E_n$ algebras, we should only expect $A$ as above to be an $E_{n,2}$ algebra. So the theorem above corresponds to the statement that a homotopy $1$-type that is an $E_{1,2}$ algebra is automatically an $E_2$ algebra. Very roughly this is because of the Eckman-Hilton argument, which says that the two multiplications agree and so the second multiplication is $A_\infty$ and not just $A_2$.