Relative Turán density of ascending uniform hypergraph paths

Determine the relative Turán density \(\mathfrak{o}(P^{r}_2)\) of the ascending \(r\)-uniform path of length two for every \(r\ge 3\), and in particular determine whether \(\mathfrak{o}(P^{3}_2)=1/2\).

Background

The definition of relative Turán density extends to hypergraphs. The paper cites known lower bounds for the ascending $r$-uniform path of length two and notes a quantitative improvement for $r=4$. The exact value remains unresolved for general $r$, with the case $r=3$ singled out as a specific question; the authors also mention the analogous problem for longer paths.