How to check whether a route(URL) is an asset route?

I have a session variable in a BaseController's constructor to store current route. The code is like this:

if (!request()->ajax()) {
      session()->put('current_url', request()->fullUrl());
}
                

But sometimes, I got asset URL like this in the session value:

https://laravel.dev/favicon.ico

I checked url()->full() but its implementation is the same:

/**
     * Get the full URL for the current request.
     *
     * @return string
     */
    public function full()
    {
        return $this->request->fullUrl();
    }

How can I get the current full URL (exclude asset routes) or how can I know whether a route is an asset route or not?

Thanks!