Fritzsche–Külshammer–Reiche character-theoretic formula for the base size of ${\rm S}_{n,k}$

Let $n\geq 2k$, let $G={\rm S}_{n,k}$, and let $H$ be a point stabiliser of $G$. Write $b(G)$ for the base size of the permutation group $G$, let $\mathcal{K}(G,H)$ be the Külshammer graph of $G$ with respect to $H$, let $\operatorname{Diam}(\mathcal{K}(G,H))$ denote its diameter, and let $1_H$ and $\operatorname{sgn}\!\downarrow_H$ denote respectively the trivial character of $H$ and the restriction to $H$ of the sign character. If $d(\alpha,\beta)$ denotes graph distance in $\mathcal{K}(G,H)$, then Fritzsche–Külshammer–Reiche's conjecture.

$$b(G)=\operatorname{Diam}(\mathcal{K}(G,H))+1=d(1_H,\operatorname{sgn}\!\downarrow_H)+1.$$

This conjecture proposes an alternative character-theoretic formula for the base size in terms of the Külshammer graph; the cited work of Fritzsche, Külshammer, and Reiche is presented as the source of the conjecture, and no resolution is given here.