Closure of biased separable Brownian permutons under independent composition

Determine whether the product of two independent biased separable Brownian permutons can belong to the same one-parameter family; in particular, establish or refute the conjecture that the independent product family $\mathcal{S}\mathcal{S}$ is disjoint from $\mathcal{S}$, where $\mathcal{S}=\{\mu^p:p\in(0,1)\}$.

Background

The paper proves that the family of biased separable Brownian permutons is not closed under independent products by showing that at least one parameter value produces a product whose distribution lies outside the family. It then states the stronger conjectural possibility that every product of two members lies outside the family. The authors explain that proving this would require sufficient understanding of convolutions of the associated density functions $\alpha_p$.