Craig–Pun's log-concavity and higher-order Turán conjecture for distinct partitions

Let $q(n)$ denote the number of partitions of $n$ into distinct parts. Log-concavity means $q(n)^2\geq q(n-1)q(n+1)$, and the higher-order Turán inequalities are the corresponding higher-order inequalities for this sequence.

Craig–Pun's conjecture. The function $q(n)$ is log-concave for $n\geq33$ and satisfies the higher-order Turán inequalities for $n\geq121$.

The paper presents this as a conjecture of Craig and Pun. Its status is not resolved in the supplied material, although the paper notes that the higher-order inequalities hold for sufficiently large $n$.