Mubayi–Zhao's rational codegree Turán density conjecture for single hypergraphs

Let $k$-graphs be $k$-uniform hypergraphs, let $\beta(F)$ denote the codegree Turán density of a $k$-graph $F$, and let $\beta(\rho)$ denote the corresponding density for a family $\rho$ of $k$-graphs. Mubayi–Zhao's conjecture. Fix $k 3$. For every rational number $\alpha\in [0,1)$, there exists a $k$-graph $F$ such that $\gamma(F)=\alpha$. Only a few sporadic examples of $3$-graphs with non-zero codegree Turán density are known, so the conjecture remains open; the preceding theorem establishes the analogous statement for finite families of $k$-graphs.