Baber–Johnson–Talbot triangle-density conjecture for tripartite graphs

Let $R=\{(\alpha,\beta,\gamma)\in[0,1]^3:\alpha\beta+\gamma>1,\alpha\gamma+\beta>1,\beta\gamma+\alpha>1\}$, and let

$$\Delta(\alpha,\beta,\gamma)=\alpha^2+\beta^2+\gamma^2-2\alpha\beta-2\alpha\gamma-2\beta\gamma+4\alpha\beta\gamma.$$

Define $R_1=\{(\alpha,\beta,\gamma)\in R:\Delta(\alpha,\beta,\gamma)\geq 0\}$ and $R_2=R\setminus R_1$. Let $T_{\min}(\alpha,\beta,\gamma)$ denote the minimum number of triangles in a tripartite graph with the stated densities, divided by $n^3$. Baber–Johnson–Talbot conjecture. If $(\alpha,\beta,\gamma)\in R_2$, then

$$T_{\min}(\alpha,\beta,\gamma)=2\sqrt{\alpha\beta(1-\gamma)}+2\gamma-2.$$

The cited Baber–Johnson–Talbot result gives only the corresponding upper bound in the source, so equality remains unresolved there. The formula would determine the minimum triangle density throughout the region $R_2$.