Decentralized party with confirmation threshold > 1

Hello team,

For a decentralized party, when setting the partyToParticipantMapping with confirmation threshold > 1, we cannot submit the transaction even through the submission node.
The docs here Decentralization - Canton Network Docs stated that:

Parties with a threshold greater than 1 in the PartyToParticipant mapping cannot submit Ledger API commands. This is because command submissions always go through a single Participant Node and there is no cryptographic link between the party’s submission and the resulting transaction

A bit confused re “there is no cryptographic link between the party’s submission and the resulting transaction”.

Say we have a decentralized party alice::<name_space_c> where name_space_c is computed from local_name_space_a generated on participant1 and local_name_space_b on participant 2 and there’s no partyToKeyMapping with a confirmation threshold of 2.

If participant1 has submission right for alice::<name_space_c>, wondering why can’t the transaction gets signed by local_name_space_a’s private key that sits on participant1 and verified by participant2 that it’s indeed signed by local_name_space_a on participant1 with local_name_space_a’s public key stored on the network?

Wondering if anyone can help to clarify above. Thanks so much in advance.