Decomposable numbers as ghost orders

Prove or disprove that every decomposable positive integer is a ghost order, meaning that no finite group of that order has an endopure subgroup.

Background

A ghost order is defined as an integer n for which no group of order n has an endopure subgroup. The paper notes that all computationally verified entries in a listed sequence of decomposable numbers are ghost orders.

The authors conjecture that this phenomenon holds for every decomposable number. Establishing it would imply, together with known results that essentially all odd numbers are decomposable, that essentially all odd numbers are ghost orders.