How to sync NextCloud (or OwnCloud) with Mutt, keeping local cache?

I have a NectCloud server which I sync with my Android phone (it runs Cyanogenmod, and I don't sync with Google).

So -- I use mutt for writing email on my desktop, and I'd like to sync mutt aliases with my phone and NextCloud contacts, but keeping a local cache because I want to be able to use the aliases even when my inteet connection is down (or slow).

Is there an easy way to do this? Or maybe a relatively easy way, like setting up another NextCloud server on the desktop, and have it communicate with my already running server? (Does NextCloud support this?)

Ideally, I'd have

Mutt <--query/add--> some-local-cache <--CardDav sync--> NextCloud server

One possible complication: my desktop is behind NAT.

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