  All arithmetic non-compact ball quotients by Deligne-Mostow's unitary
monodromy group arise as sub-ball quotients of either of two spaces called
ancestral cases, corresponding to Gaussian or Eisenstein Hermitian forms
respectively. In our previous paper, we investigated the compactifications of
the Gaussian Deligne-Mostow variety. Here we work on the remaining case, namely
the ring of Eisenstein integers, which is related to the moduli space of
unordered 12 points on $\mathbb P^1$. In particular, we show that Kirwan's
partial resolution of the moduli space is not a semi-toroidal compactification
and Deligne-Mostow's period map does not lift to the unique toroidal
compactification. We give two interpretations of these phenomena in terms of
the log minimal model program and automorphic forms. As an application, we
prove that the above two compactifications are not (stacky) derived equivalent,
as the $DK$-conjecture predicts. Furthermore, we construct an automorphic form
on the moduli space of non-hyperelliptic curves of genus 4, which is isogenous
to the Eisenstein Deligne-Mostow variety, giving another intrinsic proof,
independent of lattice embeddings, of a result by Casalaina-Martin, Jensen and
Laza.
