The local weak limit conjecture for random intersection graphs

Construct the locally finite connected rooted random graph $G^\bullet$ from the partially labeled rooted random tree $T^\bullet$ as described in the source: even-generation vertices have offspring distribution $\operatorname{Poi}(\lambda)$, odd-generation vertices carry independent $X$-biased labels $(X^*,Q^*)$ and have $X^*-1$ children, and edges between even-generation vertices are retained according to the independent Bernoulli variables attached to odd-generation layers. Local weak limit conjecture. Under sufficient regularity, $G^\bullet$ is a local weak limit of the random intersection graph model. This conjecture identifies the rooted tree construction as the local limit of the model; the source does not specify the regularity assumptions or give a resolution.