Rational values and non-principality for $\ell$-degree Turán densities of hypergraphs

Let $k>\ell\ge 1$ be integers. For a family $\mathcal{F}$ of $k$-uniform hypergraphs ($k$-graphs), the $\ell$-degree Turán density $γ^{(k)}_\ell(\mathcal{F})$ of $\mathcal{F}$ is defined as the asymptotic maximum of the normalized minimum $\ell$-degree over all $\mathcal{F}$-free $k$-graphs. In th