The central-element conjecture for smash operations

Let $\mathscr{C}$ be an operad with a basepoint, let $k\geq 2$, let $l\geq 1$, and let $Y$ be a $\mathscr{C}$-space. Write $\mathcal{Z}_k[S^l,\mathscr{C}]$ for the relevant $k$-th center, and let $e_k$ denote the basepoint in arity $k$. Central-element conjecture. If $\alpha\in\mathcal{Z}_k[S^l,\mathscr{C}]$ and $\alpha(*)\sim e_k$, then

$$\theta_\alpha\simeq\mu'_k:\mathrm{FW}(S^l\times Y^k)\to Y.$$

This conjecture gives a sufficient condition for the homotopies on the faces to glue over the fat wedge, thereby producing smash operations. The source does not state a resolution.