The p-spectral Turán conjecture for k-chromatic uniform hypergraphs

Let $G$ be a $k$-chromatic $r$-graph of order $n$, where $k\geq 2$ and $n>(r-1)k$. For $p\geq 1$, write $\lambda^{(p)}(G)$ for the $p$-spectral radius of $G$, and let $Q_k^r(n)$ denote the complete $k$-chromatic $r$-graph of order $n$. The p-spectral Turán conjecture. For every $p\geq 1$,

$$\lambda^{(p)}(G)<\lambda^{(p)}\left(Q_k^r(n)\right),$$

unless $G=Q_k^r(n)$. This conjecture proposes the extremal $p$-spectral radius among $k$-chromatic $r$-graphs in the range $n>(r-1)k$, extending the preceding theorem from $3$-graphs; the generalization was posed because the authors' methods do not handle this range.