Strong worst-case hardness conjecture for counting k-cliques

Let $s$ be the uniformity of a hypergraph on $n$ vertices, and let $k$ be a fixed clique size. The problem \textsc{\#(k,s)-clique} asks for the number of $k$-cliques in the hypergraph. Let $\omega$ denote the fast matrix multiplication constant.

Strong worst-case clique-counting hardness conjecture. Any randomized algorithm $A$ for \textsc{\#(k,s)-clique} with error probability less than $1/3$ takes time $\tilde{\Omega}(n^{\omega \lceil k/3\rceil})$ in the worst case if $s=2$, and $\tilde{\Omega}(n^k)$ in the worst case if $s\geq 3$.

This stronger conjecture asserts optimality, up to polylogarithmic factors, of the best known algorithms for counting $k$-cliques in graphs and hypergraphs. It strengthens the basic $n^{\Omega(k)}$ counting-hardness conjecture and remains open.