Unimodality for the two unresolved two-parameter families

Show that q-rious unimodality holds for the two-parameter families of q-factorial ratios associated with the pairs in (5b) and (5c).

Background

The paper identifies two specific nontrivial two-parameter families for which the authors have not established the q-rious unimodality conjecture. These families are the q-analogues of super Catalan-type and related factorial ratios introduced earlier as (5b) and (5c).

The lack of a combinatorial or representation-theoretic interpretation is cited as one reason that proving unimodality for these families remains difficult. The statement is more specific than the general conjecture because it isolates two concrete families still lacking proofs.