Fixing speaker labels in a live tab transcriber after questions were misattributed

A developer building Transcribe Video to Text, a Chrome extension that transcribes video playing in a browser tab, found that questions from a host were being labeled with a guest's name in a demo interview. The problem turned out to involve two separate bugs, one on the server and one in the side panel, along with a limitation of live speaker diarization. The extension captures tab audio in Manifest V3 by splitting the work across the service worker, an offscreen document, and MediaRecorder streaming over a WebSocket. The server relays audio to Deepgram, which tags each word with a speaker number, but the original code assigned the first speaker found to an entire message. This caused mixed-speaker messages to be attributed to a single person, and the fix splits each final message into runs of consecutive words by the same speaker.