Lagrangian densities of some $3$-uniform hypergraphs

The Lagrangian density of an $r$-uniform hypergraph $H$ is $r!$ multiplying the supremum of the Lagrangians of all $H$-free $r$-uniform hypergraphs. For an $r$-uniform graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge r!\lambda{(K_{t-1}^r)}$. We say that an $r$-uniform hypergraph $H$ with $t$ vertices is $\lambda$-perfect if $\pi_{\lambda}(H)= r!\lambda{(K_{t-1}^r)}$. A theorem of Motzkin and Straus implies that all $2$-uniform graphs are $\lambda$-perfect. It is interesting to understand what kind of hypergraphs are $\lambda$-perfect. The property `$\lambda$-perfect' is monotone in the sense that an $r$-graph obtained by removing an edge from a $\lambda$-perfect $r$-graph (keep the same vertex set) is $\lambda$-perfect. It's interesting to understand the relation between the number of edges in a hypergraph and the `$\lambda$-perfect' property. We propose that the number of edges in a hypergraph no more than the number of edges in a linear hyperpath would guarantee the `$\lambda$-perfect' property. We show some partial result to support this conjecture. We also give some partial result to support the conjecture that the disjoint union of two $\lambda$-perfect $r$-uniform hypergraph is $\lambda$-perfect. We show that the disjoint union of a $\lambda$-perfect $3$-graph and $S_{2,t}=\{123,124,125,126,...,12(t+2)\}$ is perfect. This result implies the earlier result of Heftz and Keevash, Jiang, Peng and Wu, and several other earlier results.