Schelp's dense-subgraph conjecture for Ramsey paths

Let $P_k$ denote the path with $k$ vertices, and let $G$ be a graph on $3n-1$ vertices. Schelp's conjecture. If $n$ is sufficiently large and

$$\delta(G)>3|V(G)|/4,$$

then $G$ arrows $P_{2n}$, meaning that every $2$-edge-coloring of $G$ contains a monochromatic copy of $P_{2n}$. This is the dense-host analogue of the equality $R(P_{2n},P_{2n})=3n-1$; the supplied text says that only an asymptotic version had been proved at this point.