The clique-density construction conjecture for induced 4-cycles

For a real number $x\in[1/2,1]$, let $H^\star(n,x)$ be the complete multipartite graph from the clique-density construction, and let $I(C_4,x)$ denote the supremum of limiting induced $C_4$-densities among graph sequences with limiting edge density $x$.

Induced 4-cycle construction conjecture. For every real number $x\in[1/2,1]$,

$$I(C_4,x)=\lim_{n\to\infty}\rho\left(C_4,H^\star(n,x)\right).$$

For $x\le1/2$, the paper already determines $I(C_4,x)=3x^2/2$ using bipartite constructions. The conjecture proposes the corresponding clique-density construction for the remaining range $x\ge1/2$; the supplied status evidence leaves it open.