The Akbulut cork is not universal

We exhibit infinitely many exotic pairs of simply-connected, closed $4$-manifolds not related by any cork of the infinite family $W_n$ constructed by Akbulut and Yasui whose first member is the Akbulut cork. In particular, the Akbulut cork is not universal. Moreover we show that, in the setting of manifolds with boundary, there are no $\partial$-universal corks, i.e. there does not exist a cork which relates any exotic pair of simply-connected $4$-manifolds with boundary.