Conjecture for the Turán density of the tight 5-cycle with one edge removed

Let $\mathcal{C}_5^3$ be the $3$-uniform tight cycle on $5$ vertices, and let $\mathcal{C}_5^{-}$ be the $3$-uniform hypergraph obtained from $\mathcal{C}_5^3$ by removing one edge. Let $\pi(\mathcal{C}_5^{-})$ denote its Turán density. Conjecture for $\mathcal{C}_5^{-}$.

$$\pi(\mathcal{C}_5^{-})=\frac{1}{4}.$$

The paper identifies this as a less well-known open problem concerning a specific $3$-uniform hypergraph; its exact Turán density is not determined there.