The same way that message boards do this already on Freenet.

See https://freenetproject.org/understand.html.

Something like:

Publish a site using a USK (perhaps one per recipient), containing all encrypted messages sent recently. Update the site with an additional message when you need to send one. The recipient checks it for updates periodically.

I'm not sure if my understanding is inaccurate, but since Frost already achieves something like this, I don't see that it's automatically impossible such as what you seem to be inferring.

In fact, now that I look, there's: https://freenetproject.org/freemail.html

I'm dubious about the safety of using non-Freenet clients, but surely this demonstrates that it's possible to build a messaging system built over Freenet?