Parity-density conjecture for partitions with odd multiplicities

Let $a(n)$ denote the number of partitions of $n$ in which every part occurs with odd multiplicity. For a sequence, say that its odd values have density $d$ when the corresponding limiting proportion exists. Parity-density conjecture. The coefficients $a(8m+7)$ are odd with density $1/2$. Equivalently, $a(n)$ is odd with density $1/16$. This concerns the arithmetic progression $n\equiv 7\pmod{8}$, the only progression not covered by the paper's preceding parity characterizations; computational data suggests that its behavior differs sharply from the density-zero behavior outside this progression, but the conjecture remains wide open.