  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.
