Improve chat widget cross-tab synchronization for subscription events

[coderabbitai]

Problem Description

The chat widget uses the Broadcast Channel API to notify other browser tabs when a user has subscribed (submitted the form) to start the chat. However, there's a race condition where other tabs could reload before the chatbot responds with the initial "Get Started" message.

Current Implementation

Currently, we use a timeout workaround in the sendGetStarted function:

// Notify other tabs that we've subscribed and sent the 1st msg (so they reload)
setTimeout(() => {
  postMessage({ event: "subscribed" });
}, 500);

Issue

The 500ms timeout is arbitrary and doesn't guarantee that the chatbot has responded with the initial message. This can lead to:

• Race conditions where tabs reload too early
• Inconsistent user experience across tabs
• Potential loss of the initial bot response

Desired Solution

Find a better way to synchronize cross-tab communication that doesn't rely on arbitrary timeouts. Possible approaches:

• Wait for the actual bot response before broadcasting
• Use a more sophisticated event-driven approach
• Implement proper state synchronization mechanisms

Context

This affects the chat widget's multi-tab functionality and user experience consistency.

References

• PR: fix: ws connection mutex #1356
• Comment: fix: ws connection mutex #1356 (comment)
• Requested by: @marrouchi