Convergence of two-point Pad\'e approximants to piecewise holomorphic functions

Let $f_0$ and $f_\infty$ be formal power series at the origin and infinity, and $P_n/Q_n$, with $\mathrm{deg}(P_n),\mathrm{deg}(Q_n)\leq n$, be a rational function that simultaneously interpolates $f_0$ at the origin with order $n$ and $f_\infty$ at infinity with order $n+1$. When germs $f_0,f_\infty$ represent multi-valued functions with finitely many branch points, it was shown by Buslaev that there exists a unique compact set $F$ in the complement of which the approximants converge in capacity to the approximated functions. The set $F$ might or might not separate the plane. We study uniform convergence of the approximants for the geometrically simplest sets $F$ that do separate the plane.