Does the following hold?

For every bipartite graph $H$ and every graph $G$ with $e(G)\geq 0.1(v(G))^2$, 
    $$t(H,G)\geq t(K_2, G)^{e(H)}.$$

If not sure, is this a equal question as Sidorenko's conjecture or a subquestion of Sidorenko?