Exact minimum triangle count in the second density region

Prove that for every density triple (α,β,γ) in the region R_2, the minimum normalized number of triangles in a tripartite graph with those edge densities equals 2√(αβ(1−γ))+2γ−2.

Background

The paper invokes a result of Baber, Johnson, and Talbot describing the minimum triangle density for tripartite graphs with prescribed pairwise edge densities. In the region R_1, the exact value is known to be α+β+γ−2.

For the complementary region R_2, the cited result supplies only an upper bound given by 2√(αβ(1−γ))+2γ−2. The cited conjecture asserts that this upper bound is in fact the exact minimum, leaving the corresponding extremal problem unresolved.