Conductors and minimal discriminants of hyperelliptic curves with rational Weierstrass points

Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $\mathop{\textrm{char}} k \neq 2$. Assume that the Weierstrass points of $C$ are $K$-rational. Let $S = \mathop{\textrm{Spec}} R$. Let $\mathcal{X}$ be the minimal proper regular model of $C$ over $S$. Let $\mathop{\textrm{Art}} (\mathcal{X}/S)$ denote the Artin conductor of the $S$-scheme $\mathcal{X}$ and let $\nu (\Delta)$ denote the minimal discriminant of $C$. We prove that $-\mathop{\textrm{Art}} (\mathcal{X}/S) \leq \nu (\Delta)$. As a corollary, we obtain that the number of components of the special fiber of $\mathcal{X}$ is bounded above by $\nu(\Delta)+1$.