Like toric varieties, toric hyperkähler varieties have a combinatorial construction and an abstract characterisation. Whether the classes of varieties arising combinatorially/abstractly are equal is still conjectural. Proudfoot has divided the proof into two conjectures. Recent work of Namikawa has made much progress on the first conjecture, which deals with affine toric hyperkähler varieties. In this talk we discuss recent joint work with Kaplan, Schedler, and Proudfoot on the second conjecture, which states that all crepant partial resolutions of ‘standard’ affine toric hyperkähler varieties arise combinatorially via tilings of zonotopes.