The Whitehead conjecture that any subcomplex of an aspherical 2-complex is itself aspherical, ie, that asphericity is hereditary. In studying this property, several different "flavors" of asphericity have been examined. Cohen-Lyndon asphericity (CLA) and diagrammatic reducibility (DR) are both known to be hereditary. It has been long-known that CLA does not imply DR via the dunce cap, but whether or not DR implied CLA was unclear. Indeed, many classes of DR presentations are also CLA; staggered, cyclically reduced presentations, as well as presentations satisfying the hyperbolic weight test are both DR and CLA. We resolve the logical independence of DR and CLA by giving a presentation of the integers which is DR, but not CLA.