The exceptional-index conjecture for Stirling numbers S(n,5)

For $n\in\mathbb{N}$, let $q_n=\nu_{2}(S(n,5))$, and call $i$ exceptional when $q_{4i}\neq q_{4i+3}$. The observed exceptional indices begin with $7,39,71,103,\ldots$. The exceptional-index conjecture. The set of exceptional indices is

$$\{32j+7:j\geq 1\}.$$

This is a conjectural refinement of the analysis of the two non-constant residue classes for $S(n,5)$; the supplied text gives no proof or resolution.