[BUG] dcc.Patch() wipes persistence of sibling pattern-matching elements

Sibling of: https://github.com/plotly/dash/issues/3681
Environment

dash                 4.4.0

When patching a new element to list of elements with pattern matched ids, persistence is wiped from all sibling IDs.

Minimal example:

from dash import ALL, Dash, Input, Output, Patch, dcc, html, clientside_callback, callback

app = Dash(__name__)

def make_input(index):
    return dcc.Input(
        id={"type": "inp", "index": index},
        value="initial",
        persistence=True,
        persistence_type="local",
    )


app.layout = html.Div(
    [
        html.Button("add", id="add"),
        html.Button("clear", id="clear"),
        html.Div([make_input(0), make_input(1)], id="container"),
        html.Div(id="display"),
        html.Pre(id="storage"),
        dcc.Interval(id="local_storage_poll", interval=250),
        dcc.Input("Hello, I don't wipe", id='non-pattern-matching-id', persistence=True, persistence_type='local')
    ]
)


@callback(
    Output("container", "children"),
    Input("add", "n_clicks"),
    prevent_initial_call=True,
)
def add_input(n):
    patch = Patch()
    patch.append(make_input(n + 1))
    return patch


@callback(Output("display", "children"), Input({"type": "inp", "index": ALL}, "value"))
def show(values):
    return "|".join(values)


# Debug info, show local storage to make it easier to noticed when it's wiped
clientside_callback(
    """
    function display_persisted_vals() {
        return Object.keys(window.localStorage)
            .filter(k => k.startsWith('_dash_persistence.'))
            .map(k => k + ' = ' + window.localStorage.getItem(k))
            .join('\\n') || '(nothing persisted)';
    }
    """,
    Output("storage", "children"),
    Input("local_storage_poll", "n_intervals"),
)

# Debug helper, clear local storage to do a clean test
# (helpful after the fix when the storage doesn't wipe constantly, lol)
clientside_callback(
    """
    function clear_persistence() {
        Object.keys(window.localStorage)
            .filter(k => k.startsWith('_dash_persistence.'))
            .forEach(k => window.localStorage.removeItem(k));
        window.location.reload();
        return window.dash_clientside.no_update;
    }
    """,
    Output("clear", "n_clicks"),
    Input("clear", "n_clicks"),
    prevent_initial_call=True,
)


if __name__ == "__main__":
    app.run(debug=True)

Here is a run of the minimal example app

Persistence works without a patch.
When a patch is applied persistence is wiped from siblings, but not from unrelated elements.

Beginne damit, die minimale Dash-Anwendung im Issue mit Dash 4.4.0 auszuführen. Verfolge anschließend, wie dcc.Patch() children mit pattern-matching IDs aktualisiert und wie persistence wiederhergestellt wird. Als abgeschlossen gilt die Aufgabe, wenn das Anhängen eines neuen Geschwisters die lokale persistence für vorhandene pattern-matched inputs bewahrt, während eine nicht zusammenhängende persistence unverändert bleibt.