The improved ratio conjecture for vertex-disjoint triangles in tripartite graphs

Let $k\geq 2$ and let $n$ be sufficiently large. For $1\geq\alpha,\beta,\gamma\geq 0$, an $(k,n)$-cyclic triple is a triple satisfying the six cyclic density inequalities stated in the paper, and the main theorem asserts that such densities force $k$ vertex-disjoint triangles when $n\geq 5k+2$. Improved ratio conjecture. The bound $n\geq 5k+2$ can be improved to $n>Ck$ for any constant $C>1$, for sufficiently large $n$. This would substantially improve the range in which the cyclic density conditions guarantee $k$ vertex-disjoint triangles; the source presents it as a belief and gives no resolution.