Almost-sure convergence conjecture for non-leaf-free multicyclic subgraph counts

Fix a connected multicyclic graph $H$ with $2$-core $H'$, and let $t$ be the number of vertices in $H\setminus H'$. For a fixed $m\geqslant m_H$, let $W_n$ and $W'_n$ denote the numbers of copies of $H$ and $H'$ in $G^m_n$, respectively. Convergence conjecture for multicyclic counts. There exists a constant $c>0$ such that

$$(\log n)^{-t}W_n-cW'_n\overset{\mathrm{a.s.}}{\longrightarrow}0$$

as $n\to\infty$, and consequently

$$\frac{W_n}{\log^t n}\overset{\mathrm{a.s.}}{\longrightarrow}cW'_\infty.$$

The theorem preceding the conjecture gives the expectation and variance bounds for this family; the conjectured limit is not expected to have a simple distributional form.