They should support both, but BIP47 is outdated and not as private at BIP351.
Discussion
BIP351 is outdated and not as private as Silent Payments- https://gist.github.com/RubenSomsen/c43b79517e7cb701ebf77eec6dbb46b8
Silent Payments doesn't work with light clients such as mobile wallets. It needs full block data. The theory behind the standard is sound but in practice you need to support light clients or you won't find adoption.
BIP351 elegantly solved that problem by outsourcing notification scanning. Worst case scenario (you don't trust anyone), you run your own notification scanner. Most people will trust a few high profile notification repos or the one run by their mobile wallet vendor.
Mobile wallets still have to connect to the user's full node running at home, so there's no tradeoffs for Silent Payments.
Which mobile wallets require you to run your own node at home? Most rely either on centralized backends or some kind of SPV/Neutrino system.
BIP351 gives you the ability to run your own notification scanner if you're paranoid, but in practice most users can rely on a few high profile scanners and cross check them for honesty.
You to run a node in order to use ANY wallet. If you are not validating the chain, you are not using Bitcoin, you are merely trusting someone else to use Bitcoin for you.
Does Wasabi download and consensus validate every single block?
Furthermore, BIP351 only need notifications for the first interaction between two peers. After that, they can rely solely on SPV or Compact Block Filters in order to interact.
And they don't even need to rely on on-chain notifications since they can connect out-of-band via email, Nostr or any other communication medium.
In that sense, the featureset of 351 is strictly a superset of what SP offers and therefore superior.
Another downfall of SP is that it isn't even a standard. It's a stream of consciousness in a GitHub gist, without a real spec or reference implementation.
If you are communicating out of band, then just send the address directly. No BIP needed.
We cover all these scenarios here:
https://github.com/bitcoin/bips/blob/master/bip-0351.mediawiki#motivation