Oddness conjecture for valid hook configurations

For $n\geq 3$, let $\operatorname{VHC}(S_{n-1})$ denote the set of valid hook configurations of permutations in $S_{n-1}$. Oddness conjecture. If $n\geq 3$, then $|\operatorname{VHC}(S_{n-1})|$ is odd if and only if $n+1$ is a power of $2$. This conjecture is based on the initial values of the sequence; no resolution is given in the source.