Exact relation between the idempotent-sum-free constant and the Erdős–Burgess constant for cyclic semigroups

Let ${\rm C}_{k;n}$ be a cyclic commutative semigroup with parameters $k$ and $n$, and let ${\hat{\sl l}}({\rm C}_{k;n})$ denote the threshold associated with idempotent-sum-free sequences, while ${\sl l}({\rm C}_{k;n})$ denotes the Erdős–Burgess constant. The conjecture. For $k>n$,

$${\hat{\sl l}}({\rm C}_{k;n})={\sl l}({\rm C}_{k;n})-1.$$

The claim concerns the relationship between the auxiliary idempotent-sum-free threshold and the Erdős–Burgess constant in the cyclic case; the preceding discussion derives several exact formulas and bounds for particular parity cases, but this stated relation is not accompanied by a resolution in the supplied text.