The $4k+1$ conjecture for perfect Skolem-type sequences

A perfect Skolem-type sequence of order $n$ is a sequence constructed from an ordinary perfect Skolem set of cardinality $n$. The $4k+1$ conjecture. For every order $n$, the number of perfect Skolem-type sequences is of the form

$$4k+1$$

for some integer $k$. The conjecture is suggested by the computed counts through order $13$. The observed oddness comes from reversal pairing, with one reversal-invariant sequence; the stronger congruence modulo $4$ remains presented as a conjecture.