Nonexistence of parallel no-HGS extensions of squarefree degree

Let $L/K$ be a separable field extension of squarefree degree. Say that $L/K$ has the parallel no-HGS property if it admits a Hopf–Galois structure while a parallel extension associated with it admits no Hopf–Galois structure of any type. Squarefree-degree nonexistence conjecture. There is no separable extension of squarefree degree admitting the parallel no-HGS property. The conjecture is motivated by the absence of examples in squarefree degree, in contrast with the degree-$8$ examples constructed in the paper; its general status is not established here.