Let $A$ be a (complete) DVR with maximal ideal $m$ and residue field $A/m=k$ of positive characteristic, and let $K$ be its fraction field.
Let $X, Y$ be curves (or more generally cycles of complementary dimension) on a surface (scheme of higher relative dimension) $Z$ over $A$ (all schemes are assumed proper). Denote $X', Y', Z'$ base changes to $\mathrm{Spec} K$ (general fibre), and $X^\circ, Y^\circ, Z^\circ$ base changes to $\mathrm{Spec} k$ (special fibre). Assume $X'$ and $Y'$ meet at a point $P$. By properness, this point comes from an $R$-point on $Z$, which in turn has a reduction $P^\circ$ on $Z^\circ$.
How can one relate intersection multiplicities $i(X^\circ, Y^\circ; P^\circ)$ and $i(X', Y'; P)$? Is it true that the latter is always less than or equal than the former?