On elements of degree $g$ in the separating semigroup of a real curve of genus $g$

Let $C$ be a separating real algebraic curve of genus $g$ whose real locus consists of $r$ components and let $S=\mathrm{S}\mathrm{e}\mathrm{p}(C)$ be its separating semigroup. An element $\delta \in S$ is called extendable if $\delta +{\mathbb{N}}_0^r\subset S$, where ${\mathbb{N}}_0=\{n\in \mathbb{Z}\mid n\ge 0\}$. We prove that an element of $S$ is extendable if it is realizable by a separating mobile divisor of degree $g$ and projective rank 1.