Counter-examples to the Baker–Gammel–Wills conjecture and patchwork convergence

I review some of the by now classic conjectures concerning the pointwise convergence of the diagonal Padé approximants and the very recent counter-examples to all of them. As the counter-examples all correspond to bounded associated continued fractions (Wallʹs family of complex, bounded J -matrices), I review and extend some of the known convergence results. I propose a new conjecture which I call the patchwork conjecture, which restores uniform convergence by means of the use of a finite number of infinite sequences of diagonal Padé approximants instead of just one as in the classic conjectures.