Conjecture on the nonbipartite Ramsey number for $C_4$-free color classes

Let $r(K_n,C_4,3)$ denote the minimum number of colors in an edge-coloring of the complete graph $K_n$ with no monochromatic copy of $C_4$ in the relevant 3-color configuration.

Nonbipartite Ramsey conjecture.

$$r(K_{n}, C_4, 3) = \frac{n}{2} + O(1)$$

and

$$r(K_{n}, C_4, 3) = \frac{n-1}{2}$$

for infinitely many $n$.

The paper proves the weaker estimate $r(K_n,C_4,3)=n/2+O(n^{1-\delta})$ for some $\delta>0$. The conjecture asks whether the error term can be improved to $O(1)$ and whether the lower bound is attained infinitely often; this is related to essentially resolvable Steiner triple systems whose pairs of color classes have union of girth $5$.