Gyárfás–Rusza–Sárközy–Szemerédi conjecture for monochromatic paths in balanced tripartite graphs

Let $K_{n,n,n}$ be the complete tripartite graph with three vertex classes of size $n$. For graphs $G$ and $H_1,H_2$, write $G\mapsto(H_1,H_2)$ if every red-blue edge-coloring of $G$ contains a red copy of $H_1$ or a blue copy of $H_2$.

Gyárfás–Rusza–Sárközy–Szemerédi conjecture. For every positive integer $n$,

$$K_{n,n,n}\mapsto(P_{2n+1},P_{2n+1}).$$

This conjecture asks for the exact Ramsey bound in the complete balanced tripartite host graph. The cited work established the asymptotic bound $K_{n,n,n}\mapsto(P_{2n-o(n)},P_{2n-o(n)})$; the exact assertion remains open in the supplied source.