The Bell–Ghioca–Tucker conjecture for intersections of two étale orbits

Let $X$ be a projective variety over $\overline{\mathbb Q}$, let $H$ be an ample $\mathbb R$-divisor on $X$ over $\overline{\mathbb Q}$, and let $f,g\colon X\longrightarrow X$ be étale endomorphisms over $\overline{\mathbb Q}$ satisfying

$$f^*H\equiv\delta_fH,\qquad g^*H\equiv\delta_gH$$

in $\operatorname{NS}(X)_{\mathbb R}$ for some $\delta_f,\delta_g\in\mathbb R_{>1}$. For points $P,Q\in X(\overline{\mathbb Q})$, define

$$S_{f,g}(P,Q):=\{(m,n)\mid f^m(P)=g^n(Q)\}.$$

Bell–Ghioca–Tucker conjecture. The set $S_{f,g}(P,Q)$ is a union of finitely many sets of the form

$$\{(a_i+b_i\ell,c_i+d_i\ell)\mid \ell\in\mathbb Z_{\geq0}\}$$

for some non-negative integers $a_i,b_i,c_i,d_i\in\mathbb Z_{\geq0}$. The conjecture is known when $\delta_f=\delta_g$, but remains open in general.