  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. By virtue of topological recursion
and the Grothendieck-Riemann-Roch formula this, in turn, gives new and uniform
proofs of almost all ELSV-type formulas discussed in the literature.
