Sidorenko's conjecture is a fundamental question in combinatorics and graph theory. Here is the analytic version: A graphon is a bounded measurable symmetric function $W:[0,1]^2\to [0,1]$. For any graphon $W$ and any bipartite graph $H$ with bipartition $(A,B)$ where $|A|=v_1,\ |B|=v_2$, the density of $H$ in $W$ is defined as: $$t(H,W)=\int_{[0,1]^{v(H)}}\prod_{ij\in E(H)}W(x_i,y_j)\prod_{i\in A} dx_i \prod_{i\in B} dy_j .$$ Here is Sidorenko's conjecture and its two equal versions as I know: Sidorenko's Conjecture: For any $p\in [0,1]$ and any bipartite graph $H$ and any graphon $W$ with $\int W=p$ the following holds: $$t(H,W)\geq p^{e(H)}.$$
Sidorenko's Conjecture': For any $p\in [0,1]$ and any bipartite graph $H$ and any graphon $W$ with $\int W=p$, there exist a constant $0<c=c(H)<1$ the following holds: $$t(H,W)\geq c\cdot p^{e(H)}.$$
Sidorenko's Conjecture'': For any $p\in [0,1]$ and any bipartite graph $H$ and any regular graphon $W$ with $\int W=p$ the following holds: $$t(H,W)\geq p^{e(H)}.$$
Does the following proposition also equal Sidorenko's conjecture?
Proposition: For any $p\in [0,1]$, there exists $p'\in [p,\sqrt{p}]$ such that for any bipartite graph $H$ and any graphon $W$ with $\int W=p'$ the following holds: $$t(H,W)\geq (p')^{e(H)}.$$
