The OD-characterizability conjecture for symmetric groups

Let $S_m$ denote the symmetric group of degree $m$. A finite group is OD-characterizable if it is uniquely determined, among finite groups, by its order and degree pattern; it is $3$-fold OD-characterizable if exactly three groups share those invariants. The symmetric-group OD-characterizability conjecture. All symmetric groups $S_m$, with $m\neq 10$, are OD-characterizable or $3$-fold OD-characterizable.

The conjecture extends a proposed classification of symmetric groups by order and degree pattern. The paper gives a revised finite list and proves that $S_{27}$ is $38$-fold OD-characterizable, so the conjecture as stated is refuted.