Best way to store melons

I had to pick 5 melons today - they turned yellow all together and we having pretty rainy weather for last two and future 3 days, so I don’t want them to rot/crack. They yellow, but not too soft, so they can stay on the counter for a day or two for best eating. But I need to keep them for a week, so my husband who is traveling now can enjoy them too. So what is the best way, refrigerate them now, still not completely ripe, or let them ripe on the counter and refrigerate after that?

refrigerate them now.

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