🚀 go-pugleaf

A generalised regular family (GRF) of sets is represented by a 
vector (bot;top[1],...,top[n]) where bot and each top[i] are sets, 
and each top[i] is a superset of bot.  The GRF consists of all  
sets X such that bot <= X <= top[i] for some i, where "<=" means 
"subset".  A _simple_ GRF is one with n=1.

INSTANCE:  A GRF F; and integer K >= 0.
QUESTION:  Is F equal to the disjoint union of at most K
           simple GRFs?

What is the complexity of this problem?  I suspect it is NP-complete
but can't rule out some simple dynamic program (or something).
This problem arose from a graph computation that runs faster
if GRFs can be quickly split into a small number of simple GRFs.

Thanks,
Brendan McKay.