  We prove a recent conjecture of Borot et al. that a particular universal
closed algebraic formula recovers the correlation differentials of topological
recursion after the swap of $x$ and $y$ in the input data. We also show that
this universal formula can be drastically simplified (as it was already done by
Hock).
  As an application of this general $x-y$ swap result, we prove an explicit
closed formula for the topological recursion differentials for the case of any
spectral curve with unramified $y$ and arbitrary rational $x$.
