Your first proposal is quite close. More precisely, “Ils n’ont plus d’intimité” would be “they don’t have intimacy anymore”. If you add “même” it means “they don’t even have intimacy anymore”.
Exact translations are a bit difficult, but to add something to the picture consider the following correspondances:
- They don’t have more intimacy ↔ Ils n’ont pas plus d’intimité (qu’avant)
- They have no more than intimacy ↔ Ils n’ont rien de plus que l’intimité
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