For a well-based space $X$ denote by $C(\mathbb{R};X)$ the unordered configuration space of points on the real line with labels in $X$, and a point can vanish if its label reaches the basepoint. (Alternatively, you can think about the free $E_1$-algebra over the based space $X$.) Note that $C(\mathbb{R};X)$ is filtered by subspaces $C_{\le r}(\mathbb{R};X)$ which contain configurations of at most $r$ labelled points.

Now let $X$ and $Y$ be well-based and path-connected spaces. According to Segal, we have homotopy equivalences $$C(\mathbb{R};X\times Y)\to \Omega\Sigma(X\times Y),$$ and we use that $\Sigma(X\times Y)$ splits up to homotopy into $\Sigma X\vee \Sigma Y\vee \Sigma(X\wedge Y)$. We also have a homotopy equivalence $$C(\mathbb{R};X\vee Y\vee (X\wedge Y))\to \Omega \Sigma(X\vee Y\vee (X\wedge Y)).$$ Now my question is: Can we invert the homotopy equivalences in such a way that the resulting equivalence $$C(\mathbb{R};X\vee Y\vee (X\wedge Y)) \to C(\mathbb{R};X\times Y)$$ is filtration-preserving? Or is there even an explicit, geometric description of such a map?