Parity conjecture for the seaweed index of odd-part partitions

Let $p=(\lambda_1,\ldots,\lambda_l)$ be a partition of $n$ into odd parts, and let $\operatorname{ind}_S(p)$ denote its associated seaweed-index statistic. Parity conjecture. The index $\operatorname{ind}_S(p)$ is even if and only if $l\equiv 1,2\pmod 4$, and it is odd if and only if $l\equiv 0,3\pmod 4$. This conjecture connects the parity of a seaweed-algebra statistic with the number of parts of an odd partition; its resolution is not supplied in the source.