The H9 non-extremality conjecture in the second parameter region

Let $\alpha,\beta,\gamma\in[0,1]$, let $R_2\subseteq[0,1]^3$ be the second region defined in the paper, and let $H_9$ be the specified tripartite graph. For a weighting $w$, $(H_9,w)\in\mathbf{Tri}(\alpha,\beta,\gamma)$ means that it is a weighted tripartite graph with the prescribed edge densities. A graph is extremal if its triangle density equals $T_{\min}(\alpha,\beta,\gamma)$, and vertex minimal if no graph with the same triangle density has fewer vertices. H9 non-extremality conjecture. If $(\alpha,\beta,\gamma)\in R_2$, then every weighting $w$ such that $(H_9,w)\in\mathbf{Tri}(\alpha,\beta,\gamma)$ makes $(H_9,w)$ either non-extremal or non-vertex-minimal. The source states this as a sufficient conjecture for proving the proposed piecewise formula, with no resolution supplied.