Shalaby's strong Skolem starter conjecture

Let $n$ be an integer with $n\geq11$, and let $\mathbb{Z}_n$ be the additive cyclic group of order $n$. A strong Skolem starter is a starter for $\mathbb{Z}_n$ that is both Skolem and strong. Shalaby's conjecture. If

$$n\equiv1,3\pmod{8},$$

then $\mathbb{Z}_n$ admits a strong Skolem starter. The conjecture gives the expected existence criterion for strong Skolem starters beyond the orders for which explicit constructions were known. The supplied text does not establish the conjecture in full; the later result in the paper concerns only specified families of orders.