Determine the exact minimum triangle count in the unresolved density regime

Prove that for every density triple (α,β,γ) in the region R_2 defined by Δ(α,β,γ)<0 and αβ+γ>1, αγ+β>1, and βγ+α>1, the minimum normalized number of triangles in a tripartite graph with edge densities α, β, and γ equals 2√(αβ(1−γ))+2γ−2.

Background

The paper invokes a result of Baber, Johnson, and Talbot giving the exact minimum normalized triangle count in one region R_1 of the cyclic-density domain and an upper bound in the complementary region R_2. The normalized triangle count is denoted T_min(α,β,γ).

The conjecture asserts that the displayed upper bound is actually exact throughout R_2. Resolving it would complete the tripartite analogue of a triangle-counting extremal problem and would sharpen the connection between prescribed pairwise densities and the number of triangles.