Is there a "Cauchy-Schwarz proof" of the following inequality?

**Theorem.** Given $f \colon [0,1]^2 \to [0,1]$, one has
$$
\int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) \, dxdy\right)^3.
$$

*Background.*
This inequality is due to Blakley and Roy (1965). In fact, they proved even more, namely when the LHS corresponds to a path of length $k$ (above $k=3$) and the RHS is $(\int f)^k$.

This is a special case of a more general Sidorenko's conjecture, which claims that $t(H,W) \geq (\int W)^{e(H)}$ for any bipartite graph $H$. The general case of Sidorenko's conjecture is still open. See, e.g., this note by Conlon, Fox, and Sudakov (although there has been some other progress since then).

Szegedy and Li gives a different proof of the above inequality, using convexity of the logarithm function.

Also see the paper of Kim, Lee, and Lee for another approach.

On page 28 of Lovasz' book on graph limits, it states this inequality without proof, and then says

... and this is already quite hard, although short proofs with a tricky application of the Cauchy–Schwarz inequality are known.

So my question is: how does one prove the inequality above using Cauchy-Schwarz?

*Update:* It has been shown that there is no vanilla sum-of-squares proof of the inequality https://arxiv.org/abs/1812.08820