  Let $X$ be a smooth projective curve over a complete discretely valued field
$K$. Let $L/K$ be the minimal extension such that $X \times_K L$ has a
semi-stable model, and write $e(L/K)$ for the ramification index of $L/K$. Let
$e(X)$ be the so-called ``stabilization index'' of $X$, defined by Halle and
Nicaise as the lcm of the multiplicities of the ``principal'' irreducible
components of a minimal regular snc-model of $X$.
  It is known that if $L/K$ is tame, then $e(X) = e(L/K)$. If one drops the
tameness assumption, but instead assumes that $X$ has index one and potentially
multiplicative reduction, Halle and Nicaise ask if the equality $e(X) = e(L/K)$
still holds. We prove that $e(X)$ divides $e(L/K)$ in this situation, but we
give examples, in every residue characteristic, of $X$ with $K$-rational points
and potentially multiplicative reduction such that $e(X) \neq e(L/K)$.
