This is the context:
D’autres régions sont propices aux loyers onéreux : les destinations de vacances. Et les Grisons remportent la palme, loin devant le Valais et le Jura. Les 2000 francs de loyer y sont fréquents en de nombreux endroits.
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Literally, remporter la palme is to win a trophy.
It is also an idiom to designate a “winner” (quotation marks included). Very often it's used sarcastically, then it would of course mean “the worse”.
In your sentence it is not completely clear whether having high-priced rents is considered good or bad, it depends on the context. However it is clear that Les Grisons have the highest rates.