The conjectured value of the minimum $4$-clique density with bounded independence number

Let $f(n,k,l)$ denote the minimum possible number of copies of $K_k$ in an $n$-vertex graph with independence number less than $l$, and let $c_{k,l}$ be the corresponding asymptotic minimum density. In particular, $c_{4,4}$ concerns the minimum asymptotic density of $4$-cliques in graphs with independence number less than $4$.

Conjectured value of $c_{4,4}$.

$$c_{4,4}=\frac{-11+14\times 2^{1/3}}{192}.$$

The value is proposed as a promising case for the approach developed in the paper; the preceding discussion notes that analogous numerical predictions for other parameter pairs are made rigorous here, while this case remains conjectural.