The Tverberg admissible-prescribable problem asks for a topological relaxation of the following convex-geometric question: which dimensions can be prescribed such that in any sufficiently large point set in Euclidean $d$-space there are convex hulls of those prescribed dimensions that share a common point. The AP conjecture gives a possible characterization of those dimensions. I will construct counterexamples to the AP conjecture, showing a further distinction between results in the affine setting and their continuous generalizations. This is the first instance of such a distinction persisting asymptotically for arbitrarily large point sets.