Sigma clique covering growth conjecture for complete multipartite graphs

For positive integers $t$ and $d$, let $K_t(d)$ denote the complete $t$-partite graph with each part of size $d$. Let $f$ be a function and let $c$ be a constant.

Sigma clique covering growth conjecture. For every positive integers $t$ and $d$, if $t\geq f(d)$, then

$$\operatorname{scc}(K_t(d))\geq cd^2t\log t.$$

The conjecture would make the known upper bound for $\operatorname{scc}(K_t(d))$ sharp up to a constant factor for sufficiently large $t$. It is proved in the source for $d=2$, while the general case remains open.