Universal interior conjecture for sums of incommensurable missing digits sets

Let $A,B\subset[0,1]$ be missing digits sets with bases $p_A$ and $p_B$, and define $T_x(y)=xy$. Universal interior conjecture. Suppose that

$$\frac{\log p_A}{\log p_B}\notin\mathbb{Q}$$

and

$$\dim_{\mathrm{H}}A+\dim_{\mathrm{H}}B>1.$$

Then $A+T_x(B)$ contains a non-trivial interval for every $x\in\mathbb{R}\setminus\\{0\\}$. If $\log p_A/\log p_B\in\mathbb{Q}$, then $A+T_x(B)$ contains a non-trivial interval for every irrational $x$. This is proposed as a stronger, precise form of genericity for sums of missing digits sets; the source presents it as suspected and does not provide a proof.