Unimodality for the two-parameter families (5b) and (5c)

Show that the q-factorial ratios associated with the two-parameter families (a, b) = ((2m, n), (m, m − n, 2n)) and (a, b) = ((2m, 2n), (m, n, m + n)) satisfy the q-rious unimodality conjecture.

Background

The paper discusses the two-parameter families corresponding to q-factorial ratios B(m, n; q) and C(m, n; q). These families are known to be positive, but their stronger unimodality is not established by the methods used in the paper.

The authors note that a combinatorial or representation-theoretic interpretation is unavailable for these polynomials, and explicitly state that they have been unable to prove the conjectured unimodality for either family. The factor 1 + q may be necessary in some exceptional cases for B(m, n; q), whereas the authors expect unimodality without that factor for C(m, n; q).