General formula for root-of-unity monomial symmetric-function specializations

Determine a general formula for the integers obtained by evaluating the monomial symmetric polynomial $m_\lambda$ at the sequence $(\zeta,\ldots,\zeta^{b-1})$, where $\zeta$ is a primitive $b$th root of unity.

Background

The paper studies the $b$-bounded symmetric polynomials $h_k^{(b)}$ and their specializations at roots of unity. Proposition \ref{prop:cauchy} gives several expansions of $h_k^{(b)}$ in standard bases of symmetric polynomials, including an expansion in the elementary and monomial bases. In that expansion, the coefficients involve evaluations of monomial symmetric polynomials $m_\lambda(\zeta,\ldots,\zeta^{b-1})$ at all nontrivial powers of a primitive $b$th root of unity.

A general formula for these coefficients would make the monomial-basis expansion more explicit. The paper notes that formulas are known for some special cases, but leaves the general evaluation problem unresolved.