The triangle-maximisation conjecture for total clique counts on surfaces

Let $G$ be an $n$-vertex graph embeddable in a surface $\Sigma$. Let $C(K_3,G)$ be the number of copies of $K_3$ in $G$, let $C(G)$ be the total number of complete subgraphs of $G$, and let $C(K_3,\Sigma,n)$ and $C(\Sigma,n)$ be the respective maxima over all $n$-vertex graphs embeddable in $\Sigma$.

Triangle-maximisation conjecture. If

$$C(K_3,G)=C(K_3,\Sigma,n),$$

then

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

The claim formalises the proposed equivalence between maximising the number of triangles and maximising the total number of complete subgraphs. It has been verified for the surfaces listed in the paper, but is open for general surfaces.