Log-concavity conjecture for reciprocal subsum denominators

Let $\mathrm{den}(n,x)$ be the denominator polynomial associated with sums of reciprocals of subsum polynomials over partitions of $n$. Log-concavity conjecture. The sequence of coefficients of $\mathrm{den}(n,x)$ is log-concave except when $n\in\{3,5,6,7\}$. The claim is presented with experimental evidence and remains open.