Small-energy majorization conjecture on the bi-tree

Let $f,g:T^2\to\mathbb R_+$, where $g$ is superadditive in each variable and

$$\operatorname{supp} f\subset\{\mathbb I g\leq\delta\}.$$

Let $\lambda\geq 10\delta$. Small-energy majorization conjecture. There exists $\varphi:T^2\to\mathbb R_+$ such that $\mathbb I\varphi\geq\mathbb I f$ on

$$\{2\lambda\leq\mathbb I g\leq 4\lambda\},$$

and

$$\int_{T^2}\varphi^2\leq C\frac{\delta}{\lambda}\int_{T^2}f^2.$$

This is the proposed analogue on the bi-tree of the established majorization theorem with small energy on a tree. The paper constructs a counterexample showing that the conjecture is false, so the asserted uniform estimate cannot hold in this generality.