Mubayi–Pikhurko–Sudakov conjecture on Turán densities of tight cycles minus one edge

Let $C_\ell^{-}$ denote the 3-uniform tight cycle of length $\ell$ with one edge removed, and let $\pi(F)$ denote the Turán density of a 3-uniform hypergraph $F$.

Mubayi–Pikhurko–Sudakov conjecture. If $\ell\geq 5$ is not divisible by three, then

$$\pi\bigl(C_\ell^{-}\bigr)=\frac{1}{4}.$$

A recursive construction gives $C_\ell^{-}$-free 3-uniform hypergraphs of density $1/4-o(1)$ for such $ll$, establishing the lower bound; the asserted matching upper bound remains open in the source.