Optimal construction conjecture for the partially directed k-triangle Turán problem

For $k\geq 2$, let $\vec T_k$ be the partially directed $k$-triangle, and let $\pi(\vec T_k,\theta)$ denote the weighted Turán density in which directed edges have weight $\theta$.

Optimal construction conjecture.

$$\pi\left(\vec T_k,(1-1/k)^{-k+1}\right)=1$$

for every $k\geq 2$.

The conjecture asserts that the construction obtained by partitioning the vertices into parts of sizes approximately $n/k$ and $(k-1)n/k$, then directing the relevant edges toward the first part, is optimal. The source says it is confirmed for $k=2$; the general case remains open.