  We find new obstructions to the desingularization of compact Einstein
orbifolds by smooth Einstein metrics. These new obstructions, specific to the
compact situation, raise the question of whether a compact Einstein
$4$-orbifold which is limit of Einstein metrics bubbling out Eguchi-Hanson
metrics has to be K\"ahler. We then test these obstructions to discuss if it is
possible to produce a Ricci-flat but not K\"ahler metric by the most promising
desingularization configuration proposed by Page in 1981. We identify $84$
obstructions which, once compared to the $57$ degrees of freedom, indicate that
almost all flat orbifold metrics on $\mathbb{T}^4/\mathbb{Z}_2$ should not be
limit of Ricci-flat metrics with generic holonomy while bubbling out
Eguchi-Hanson metrics. Perhaps surprisingly, in the most symmetric situation,
we also identify a $14$-dimensional family of desingularizations satisfying all
of our $84$ obstructions.
