Mubayi–Rödl conjecture for the Turán density of the tight 5-cycle

Let $\mathcal{C}_5^3$ be the $3$-uniform tight cycle of length $5$, with vertex set $\{1,\dots,5\}$ and edges of the form $\{x,x+1,x+2\pmod 5\}$. Let $\pi(\mathcal{C}_5^3)$ denote its Turán density. Mubayi–Rödl conjecture.

$$\pi(\mathcal{C}_5^3)=2\sqrt{3}-3.$$

The conjecture is attributed to Mubayi and Rödl and is presented as a central open problem on tight-cycle Turán densities; the paper proves results for sufficiently long tight cycles but does not settle the length-$5$ case.