Identify a higher-dimensional analogue of connecting gonality

Determine whether there is a good analogue of connecting gonality for families of k-dimensional subvarieties, with k>1, that captures the relevant covering or irrationality properties in the way that p-connecting gonality does for families of curves.

Background

For higher-dimensional connecting families, the paper uses canonical volume as the analogue of genus and proves linear lower and upper bounds in the number p of general points. Unlike the curve case, however, no invariant corresponding to gonality is selected.

The authors mention degree of irrationality and related variants as possible candidates, but explicitly state that it is unclear whether any such notion yields a satisfactory extension of the theory for subvarieties of dimension k>1.