The simultaneous clique-maximisation conjecture for graphs on surfaces

Let $G$ be an $n$-vertex graph embeddable in a surface $\Sigma$. Let $C(G)$ denote its total number of complete subgraphs, let $C(\Sigma,n)$ be the maximum of $C(G)$ over all such graphs, and let $C(K_s,\Sigma,n)$ be the maximum number of copies of $K_s$ in an $n$-vertex graph embeddable in $\Sigma$.

Simultaneous clique-maximisation conjecture. If $C(G)=C(\Sigma,n)$, then for every $s\geqslant 0$,

$$C(K_s,G) = C(K_s,\Sigma,n).$$

This is the equality-case counterpart of the clique-count decomposition conjecture: every graph attaining the maximum total clique count should also attain each individual clique-count maximum. It has been verified for the surfaces listed in the paper, while the assertion for arbitrary surfaces remains open.