  The moduli space of complex cubic surfaces has three different, but
isomorphic, compact realizations: as a GIT quotient, as a Baily--Borel
compactification of a ball quotient, and as a compactified $K$-moduli space.
From all three perspectives, there is a unique boundary point corresponding to
non-stable surfaces. From the GIT point of view, to deal with this point, it is
natural to consider the Kirwan blowup, while from the ball quotient point of
view it is natural to consider the toroidal compactification. Both these spaces
have the same cohomology and and it is therefore natural to ask whether they
are isomorphic. Here we show that this is in fact not the case. Indeed, we show
the more refined statement that both spaces are equivalent in the Grothendieck
ring, but not $K$-equivalent. Along the way, we establish a number of results
and techniques for dealing with singularities and canonical classes of Kirwan
blowups and toroidal compactifications of ball quotients.
