The Ascoli-He-Park-Talagrand graph-decomposition conjecture at the expectation threshold

Theorem 1.1: there are absolute $k\ge2$ and $L\ge1$ such that for every $n\ge2$ and every graph $H$ on at most $n$ vertices, $E(H)=E(H_1)\sqcup\dots\sqcup E(H_k)$ with $p_c(H_i;n)\le Lq(H;n)$ for each $i$. Pieces may be empty or share vertices, and the partition depends only on $H$ and $n$. It does not give a single copy of $H$ at threshold $Lq(H;n)$ (the separate embeddings need not glue), and $k$, $L$ are not explicit.

Theorem 1.1: there are absolute $k\ge2$ and $L\ge1$ such that for every $n\ge2$ and every graph $H$ on at most $n$ vertices, $E(H)=E(H_1)\sqcup\dots\sqcup E(H_k)$ with $p_c(H_i;n)\le Lq(H;n)$ for each $i$. Pieces may be empty or share vertices, and the partition depends only on $H$ and $n$. It does not give a single copy of $H$ at threshold $Lq(H;n)$ (the separate embeddings need not glue), and $k$, $L$ are not explicit.