Use the sidebar to sync non-DM messages, just like what's done for DM
messages.
However, since sidebar messages don't say who sent a message, this
requires scanning read receipts with MSC2409: when a non-DM sidebar
message is seen, the bridge bot sends that message. Only when that
placeholder is viewed in Matrix does Puppeteer actually view the non-DM
chat to find who really sent the message. Then the bridgebot redacts its
message, and the message gets re-sent by the puppet of the LINE user who
really sent it.