Peng–Zhao Motzkin–Straus-type conjecture for hypergraphs with a large clique

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

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

Let $G$ be an $r$-graph with $m$ edges that contains a clique of order $t-1$. Peng–Zhao's Motzkin–Straus-type conjecture. Then

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

The upper bound is best possible, as the source gives an $(r,m)$-graph with one additional edge whose Lagrangian exceeds that of $[t-1]^{(r)}$. The conjecture concerns the relationship between clique number and Lagrangian and is presented as a conjecture attributed to Peng and Zhao; no resolution is supplied here.