Conjecture on intersections of independent orbit-dimension level sets

For integers $p,q\geq2$, let $T_p$ and $T_q$ be the corresponding transformations on $[0,1)$, and define

$$E_{T_p}(\alpha)=\left\{x:\dim_{\rm H}\overline{\mathcal{O}_{T_p}(x)}=\alpha\right\}.$$

Assume that $p$ and $q$ are multiplicatively independent, written $p\nsim q$, and let $\alpha_1,\alpha_2\in[0,1]$. Intersection conjecture.

$$\dim_{\rm H}\bigl(E_{T_p}(\alpha_1)\cap E_{T_q}(\alpha_2)\bigr)=\max\{\alpha_1+\alpha_2-1,0\}.$$

The preceding proposition establishes the corresponding upper bound, so the conjecture concerns the matching lower bound; the supplied text gives no resolution status.