Turán-density conjecture for partially directed k-graphs

Fix $k\geq 2$. A partially directed $k$-graph is a $k$-uniform hypergraph whose edges may be undirected or directed; let $\vec T_k$ denote the partially directed $k$-triangle. Let $\alpha\binom{n}{k}$ and $\beta\binom{n}{k}$ be the numbers of undirected and directed edges, respectively.

Partially directed triangle conjecture. For all sufficiently large $n$, every $n$-vertex partially directed $k$-graph with $\alpha\binom{n}{k}$ undirected edges and $\beta\binom{n}{k}$ directed edges that does not contain $\vec T_k$ as a subgraph satisfies

$$\alpha+(\log_2 3)\beta\leq 1.$$

The source states that this conjecture would imply the Bollobás–Brightwell–Leader counting conjecture for each fixed $k$. Its general status is not resolved in the supplied text.