= Solution
A sufficient condition is that both collections are <Alexander systems>: within each collection the essential simple closed curves are pairwise nonisotopic, are in pairwise <minimal position of curves or arcs>, have no triple intersection points, and no three curves intersect pairwise. If $\alpha_i$ is isotopic to $\beta_i$ for every $i$, the simultaneous-isotopy lemma for Alexander systems gives an ambient isotopy $H_t$ with $H_1(\alpha_i)=\beta_i$ for all $i$. Successive applications of the <bigon criterion> prove the lemma while preserving the curves already matched.
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