  The moduli space of $8$ points on $\mathbb{P}^1$, a so-called ancestral
Deligne-Mostow space, is, by work of Kond\={o}, also a moduli space of K3
surfaces. We prove that the Deligne-Mostow isomorphism does not lift to a
morphism between the Kirwan blow-up of the GIT quotient and the unique toroidal
compactification of the corresponding ball quotient. Moreover, we show that
these spaces are not $K$-equivalent, even though they are natural blow-ups at
the unique cusps and have the same cohomology. This is analogous to the work of
Casalaina-Martin-Grushevsky-Hulek-Laza on the moduli space of cubic surfaces.
The moduli spaces of ordinary stable maps, that is the Fulton-MacPherson
compactification of the configuration space of points on $\mathbb{P}^1$, play
an important role in the proof. We further relate our computations to new
developments in the minimal model program and recent work of Odaka. We briefly
discuss other cases of moduli space of points on $\mathbb{P}^1$ where a similar
behaviour can be observed, hinting at a more general, but not yet fully
understood phenomenon.
