  In this paper are discussed the results of new numerical experiments on zero
distribution of type I Hermite-Pad\'e polynomials of order $n=200$ for three
different collections of three functions $[1,f_1,f_2]$. These results are
obtained by the authors numerically and do not match any of the theoretical
results that were proven so far. We consider three simple cases of multivalued
analytic functions $f_1$ and $f_2$, with separated pairs of branch points
belonging to the real line. In the first case both functions have two
logarithmic branch points, in the second case they both have branch points of
second order, and finally, in the third case they both have branch points of
third order.
  All three cases may be considered as representative of the asymptotic theory
of Hermite-Pad\'e polynomials. In the first two cases the numerical zero
distribution of type I Hermite-Pad\'e polynomials are similar to each other,
despite the different kind of branching. But neither the logarithmic case, nor
the square root case can be explained from the asymptotic point of view of the
theory of type I Hermite-Pad\'e polynomials.
  The numerical results of the current paper might be considered as a challenge
for the community of all experts on Hermite-Pad\'e polynomials theory.
