Connectedness of self-affine sets with product digit sets

Let $T(A,\mathcal{D})$ be a self-affine set generated by an expanding matrix $A=\left[\begin{array}{rr}p& 0\\ -a& q\end{array}\right]$ and a product digit set $\mathcal{D}=\{0,1,\dots ,m-1\}\times \{0,1,\dots ,n-1\}$. We provide a necessary and sufficient condition for the $T(A,\mathcal{D})$ to be connected, which generalizes the known results.