  Single Hurwitz numbers enumerate branched covers of the Riemann sphere with
specified genus, prescribed ramification over infinity, and simple branching
elsewhere. They exhibit a remarkably rich structure. In particular, they arise
as intersection numbers on moduli spaces of curves and are governed by the
topological recursion of Chekhov, Eynard and Orantin. Double Hurwitz numbers
are defined analogously, but with prescribed ramification over both zero and
infinity. Goulden, Jackson and Vakil have conjectured that double Hurwitz
numbers also arise as intersection numbers on moduli spaces.
  In this paper, we repackage double Hurwitz numbers as enumerations of
branched covers weighted by certain monomials and conjecture that they are
governed by the topological recursion. Evidence is provided in the form of the
associated quantum curve and low genus calculations. We furthermore reduce the
conjecture to a weaker one, concerning a certain polynomial structure of double
Hurwitz numbers. Via the topological recursion framework, our main conjecture
should lead to a direct connection to enumerative geometry, thus shedding light
on the aforementioned conjecture of Goulden, Jackson and Vakil.
