Given two graphs $H$ and $G$, the homomorphism density $t(H, G)$ is defined as the proportion of mappings from the vertices of $H$ to the vertices of $G$ that preserve adjacency. Formally,
$$ t(H, G) = \frac{\text{number of homomorphisms from } H \text{ to } G}{|V(G)|^{|V(H)|}}. $$ Sidorenko conjecture The Sidorenko conjecture states that for any bipartite graph $H$ with vertex set $V(H) = A \cup B$, the following inequality holds for any graph $G$:
$$ t(H, G) \geq t(K_2,G)^{e(H)}, $$
where $K_2$ is the complete graph on two vertices. This conjecture equals to assume $G$ is regular graph.
Forcing Conjecture If additionaly, $H$ is non-forest(i.e., contains at least a cycle), $t(H, G) = t(K_2,G)^{e(H)}$ if and only if $G$ is quasi-random.
My question: Could we only consider regular host graph $G$ when studing forcing conjecture? i.e. Does Forcing conjecture equals the following statement:
For any non-forest bipartite graph $H$, the following inequality holds for any regular graph $G$: $$ t(H, G) \geq t(K_2,G)^{e(H)}. $$ Moreover, $t(H, G) =t(K_2,G)^{e(H)}$ if and only if $G$ is quasi-random for any regular graph $G$.
