Zero Distribution of Hermite-Pad\'e Polynomials and Convergence Properties of Hermite Approximants for Multivalued Analytic Functions

In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class ${\mathscr L}$. The conjectures are based in part on the numerical experiments, made recently by the authors in [26] and [27].