Gir~ao, Popielarz, and Snyder's oriented Ramsey conjecture for 1-subdivisions

Let $T_k$ be the transitive tournament on $k$ vertices, and let $H_k$ be its $1$-subdivision, obtained by subdividing every arc exactly once. For an oriented graph $H$, let $\vec{r}(H)$ be the smallest integer $n$ such that every $n$-vertex tournament contains a copy of $H$. Gir~ao, Popielarz, and Snyder's conjecture. The oriented Ramsey number of $H_k$ satisfies

$$\vec{r}(H_k)=O(k^2).$$

Previously, the bound $\vec{r}(H_k)=O(k^2\log^3 k)$ was known; the conjecture asks whether the logarithmic factor can be removed.