Positivity of immaculate structure constants for a partition factor

Let $\alpha$ be a composition and let $\lambda$ be a partition. Let ${\mathfrak S}_\alpha$ and ${\mathfrak S}_\lambda$ be the corresponding immaculate basis elements, and define coefficients ${\bf c}_{\alpha,\lambda}^\beta$ by

$${\mathfrak S}_\alpha {\mathfrak S}_\lambda=\sum_\beta {\bf c}_{\alpha,\lambda}^\beta {\mathfrak S}_\beta.$$

Immaculate product positivity conjecture. The coefficients ${\bf c}_{\alpha,\lambda}^\beta$ are non-negative integers.

This asserts positivity for products in which the second factor is indexed by a partition. The statement is attributed in the source to BBSSZ2; no resolution status is supplied.