Nikiforov's conjecture on the Lagrangian of colex hypergraphs

Let $r$-graphs be finite uniform hypergraphs, let $\lambda(m,r)$ denote the maximum Lagrangian of an $r$-graph with $m$ edges, and let $\binom{x}{r}$ be the generalized binomial coefficient for real $x$. Nikiforov's conjecture. Let $r \ge 3$. If $m = \binom{x}{r}$ for some real $x$ satisfying $x \ge r-1$, then

$$\lambda(m,r) \le m x^{-r},$$

with equality if and only if $x \in \mathbb{N}$. Nikiforov proved this for $3 \le r \le 5$ and sufficiently large $m$, while the general case is not resolved.