Zero-measure conjecture for the sum of the $p=2$ value-set closure

Let $\eta(\mathcal{T}_2)\subseteq\mathbb{R}$ be the set associated with parameter $2$, let $\overline{\eta(\mathcal{T}_2)}$ denote its closure, and let $m$ denote Lebesgue measure. Define the sumset by

$$\overline{\eta(\mathcal{T}_2)}+\overline{\eta(\mathcal{T}_2)}=\{x+y:x,y\in\overline{\eta(\mathcal{T}_2)}\}.$$

Zero-measure conjecture.

$$m\left(\overline{\eta(\mathcal{T}_2)}+\overline{\eta(\mathcal{T}_2)}\right)=0.$$

This conjecture is proposed in connection with Palis's conjecture and the paper's preceding no-interior conjecture. Although the supplied status evidence says the nonlinear regular Cantor-set case of Palis's conjecture is proved, it says that the affine self-similar case remains unresolved; no resolution of this specific conjecture is supplied.