  We study the $n$-point differentials corresponding to Kadomtsev-Petviashvili
tau functions of hypergeometric type (also known as Orlov-Scherbin partition
functions), with an emphasis on their $\hbar^2$-deformations and expansions.
  Under the naturally required analytic assumptions, we prove certain higher
loop equations that, in particular, contain the standard linear and quadratic
loop equations, and thus imply the blobbed topological recursion. We also
distinguish two large families of the Orlov-Scherbin partition functions that
do satisfy the natural analytic assumptions, and for these families we prove in
addition the so-called projection property and thus the full statement of the
Chekhov-Eynard-Orantin topological recursion.
  A particular feature of our argument is that it clarifies completely the role
of $\hbar^2$-deformations of the Orlov-Scherbin parameters for the partition
functions, whose necessity was known from a variety of earlier obtained results
in this direction but never properly understood in the context of topological
recursion.
  As special cases of the results of this paper one recovers new and uniform
proofs of the topological recursion to all previously studied cases of
enumerative problems related to weighted double Hurwitz numbers.
