q-log-convexity of binomial and polynomial sequences

Prove that the sequences of polynomials specified in Conjecture 1.1—including the sequences involving binomial coefficients with powers q^{5k}, and the sequences (S_n^{(2)}(q))_{n\ge 0} and (S_n^{(3)}(q))_{n\ge 0}—are q-log-convex.

Background

The paper defines q-log-convexity by requiring every coefficient of P_{n-1}(q)P_{n+1}(q)-P_n(q)^2 to be nonnegative. It notes that a previously conjectured q-log-convexity result for a related sequence B_n(q) remains open and then formulates new q-log-convexity conjectures based on computation.