The finite spectral-set tiling conjecture in the integers

Let $A\subset\mathbb{Z}$ be finite. It tiles $\mathbb{Z}$ by translations if there is a set of translates whose copies of $A$ partition $\mathbb{Z}$. It is a spectral set if there is a family of characters of $\mathbb{Z}$ whose restrictions to $A$ form a total orthogonal family in $L^2(A)$.

Finite spectral-set conjecture. The following conditions are equivalent:

1. $A$ tiles $\mathbb{Z}$ by translations;
2. $A$ is a spectral set.

This is the finite-set formulation equivalent to Fuglede's conjecture for finite unions of unit intervals. The supplied status evidence indicates that the equivalence was proved, so it is solved.