Conjecture on positivity and integrality of the multi-backbone ancestor coefficients

For genus $g$ and the general multi-backbone case, let $\widehat b^{(g)}_{\vec k,\vec\beta}$ denote the coefficients identified with combinations of ancestor invariants, where $\vec k$ and $\vec\beta$ are the corresponding multi-index data. Multi-backbone coefficient conjecture. All coefficients $\widehat b^{(g)}_{\vec k,\vec\beta}$ are positive integers. The preceding single-backbone result establishes positivity and integrality for the coefficients $b^{(g)}_k$; the source proposes the analogous assertion for the general multi-backbone coefficients, without giving a proof or resolution.