The smash-operation conjecture for path-connected topological operads

Let $\mathscr{C}$ be a path-connected topological operad with a basepoint, let $Y$ be a $\mathscr{C}$-space, and let $\alpha\in [S^l,\mathscr{C}(k)]$ be such that all $d_i\alpha$ are trivial. Let $\mu'_k:S^l\times Y^k\to Y$ be the iterated product on $Y$ after projection to $Y^k$, and let $\theta_\alpha:S^l\times Y^k\to Y$ be the operation determined by $\alpha$. Smash-operation conjecture. The map $\theta_\alpha$ is homotopic to $\mu'_k$ on the fat wedge of $S^l\times Y^k$, and consequently $\mu'_k-\theta_\alpha$ induces a map

$$\bar{\theta}_\alpha:S^l\wedge Y^{\wedge k}\to Y.$$

This would provide smash operations generalizing the Samelson product and induce multilinear operations on homotopy groups; the source presents it as a proposed conjecture, with no resolution stated.