Exact-level-set criterion for Solyanik failure in convex density bases

Let $\mathcal{B}$ be a homothecy invariant density basis of convex sets in $\mathbb{R}^n$. Suppose that for some $\gamma>1$, for every $0<\alpha<1$ there is a set $E_{\alpha,\gamma}$ such that

$$\left|\left\{x\in\mathbb{R}^n:M_{\mathcal{B}}\chi_{E_{\alpha,\gamma}}(x)>\alpha\right\}\right|\geq\gamma|E_{\alpha,\gamma}|.$$

Exact-level-set criterion. Then there exist a set $E_\gamma$ and a constant $c(\gamma)>1$ such that

$$\left|\left\{x\in\mathbb{R}^n:M_{\mathcal{B}}\chi_{E_\gamma}(x)=1\right\}\right|\geq c(\gamma)|E_\gamma|.$$

This is presented as a sufficient criterion for proving the preceding Solyanik conjecture: it would reduce persistent failure of the limit to the existence of a set whose maximal function equals one on a quantitatively larger set. The supplied text does not prove this criterion.