Kang–Nikiforov–Yuan's explicit spectral-radius conjecture for chromatic hypergraphs

Let $G$ be a $k$-chromatic $r$-graph of order $n>(r-1)k$, where $k\geq 2$ and $r\geq 4$. For $p\geq 1$, let $\lambda^{(p)}(G)$ denote its $p$-spectral radius. Kang–Nikiforov–Yuan's explicit conjecture. For every $p\geq 1$,

$$\lambda^{(p)}(G)<r!\left(\binom{n}{r}-k\binom{n/k}{r}\right)n^{-r/p},$$

unless $k\mid n$ and $G$ is isomorphic to $Q_k^r(n)$. This is the explicit form of the preceding spectral extremal conjecture; it gives the claimed bound in terms of the number of vertices and is unresolved for general uniformity.