Pikhurko–Zhao Motzkin–Straus conjecture for hypergraphs without a large clique

Let $t$, $m$, and $r\geq 3$ be positive integers satisfying

$$\binom{t-1}{r}\leq m\leq \binom{t-1}{r}+\binom{t-2}{r-1}.$$

Let $G$ be an $r$-graph with $m$ edges that does not contain a clique of order $t-1$.

Pikhurko–Zhao conjecture. The Lagrangian of $G$ satisfies

$$\lambda(G)<\lambda([t-1]^{(r)}).$$

This complements the corresponding clique-containing assertion and gives a strict Lagrangian bound in the same edge range. The source presents it as a conjecture; its resolution is not specified in the supplied text.