[BUG] Session appears to not shared between HTTP endpoints and websockets

[lianggecm]

If you're reporting a bug, please continue with this template.

Describe the bug
Session sets in HTTP end points and websockets have different values.

Minimal Reproducible Example
Provide a minimal code snippet that reproduces the issue. This is crucial for us to understand and fix the bug quickly.

from fasthtml.common import *
#from fh_frankenui.core import *
import sys
import uuid

app, rt = fast_app(
    pico=True,
    hdrs=(
        Script(src="https://unpkg.com/htmx.org/dist/ext/ws.js"),  # Add WebSocket extension
        ),
)
@rt("/")
async def home(session):
    old_session_id = session.get('session_id', None)
    # Get a new session_id
    session['session_id'] = str(uuid.uuid4())
    return Div(P(f"Original session_id: {old_session_id}"),
               P(f"New session_id created in Main: {session['session_id']}"),
               Button("Settings", hx_get="/settings", hx_target="#settings-container"),Div(id="settings-container"),
               Form(Input(name="msg"), Button("Send to ws"), hx_ext="ws",
            ws_connect="/ws-test",ws_send=True,), Div(id="chat-container"))


@rt('/settings')
def get(session):
    session_id_in = session.get('session_id', None)
    session['session_id'] = str(uuid.uuid4())

    return Div(P(f"Session_id sent to settings: {session_id_in}"), P(f"New session_id created in settings: {session.get('session_id', None)}"), id="settings-container")
    
@app.ws('/ws-test')
async def ws(msg:str, send, session):
    session_id_in = session.get('session_id', None)
    session['session_id'] = str(uuid.uuid4())
    await send(Div(P(f"Session_id sent to ws: {session_id_in}"),P(f"New session_id set in ws: {session.get('session_id', None)}"), id="chat-container"))

Expected behavior
In this example I fist set session_id in "/". Then change the session_id in "/settings". Afterwards, I send the session to "/ws" and change the values of session_id. The Session_id sent to the "/ws" websocket should be the session_id value set in the "/settings" endpoint. However, as seen in the picture here, the session_id sent to the "/ws" is still the same as the value that was set in "/".

Further clicks on the two buttons would show that session_id sets in either endpoints doesn't exchange/share at all.

Environment Information
Please provide the following version information:

• fastlite version: 0.0.13
• fastcore version: 1.7.19
• fasthtml version: 0.10.1

Confirmation
Please confirm the following:

• [ X] I have read the FAQ (https://docs.fastht.ml/explains/faq.html)
• [ X] I have provided a minimal reproducible example
• [ X] I have included the versions of fastlite, fastcore, and fasthtml
• [ X] I understand that this is a volunteer open source project with no commercial support.

Additional context
Add any other context about the problem here.

Screenshots
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