Tidak dapat logout pengguna di firebase saat dia menutup halaman

import FirebaseVue  from './firebase/index'
import Vue                  from 'vue'
import App                  from './App.vue'
import router           from './router'
import money                                                    from 'v-money'
import Vuelidate                                            from 'vuelidate'
import { BootstrapVue, IconsPlugin,LayoutPlugin,ModalPlugin,CardPlugin,VBScrollspyPlugin,DropdownPlugin, TablePlugin}   from 'bootstrap-vue'
import { library }                                      from '@fortawesome/fontawesome-svg-core'
import { FontAwesomeIcon }                      from '@fortawesome/vue-fontawesome'
import { VueMaskDirective }                     from 'v-mask'
import { 
    faTimesCircle,
    faPlusCircle,
    faCoffee,
    faHome,
    faLogout,
    faSignOutAlt,
    faEdit,
    faUpload,
    faDollarSign,
    faTimes,
    faClock,
    faCheck,
    faChevronLeft,
    faChevronRight,
    faArrowCircleLeft,
    faArrowCircleRight,
} from '@fortawesome/free-solid-svg-icons'

library.add(faTimesCircle,faPlusCircle,faCoffee,faHome,faSignOutAlt,faEdit,faUpload,faDollarSign,faTimes,faClock,faCheck,faChevronLeft,faChevronRight,faArrowCircleLeft,faArrowCircleRight)

Vue.component('font-awesome-icon', FontAwesomeIcon)
Vue.use(BootstrapVue)
Vue.use(IconsPlugin)
Vue.use(LayoutPlugin)
Vue.use(ModalPlugin)
Vue.use(CardPlugin)
Vue.use(VBScrollspyPlugin)
Vue.use(DropdownPlugin)
Vue.use(TablePlugin)
Vue.use(FirebaseVue)
Vue.use(require('vue-moment'));
Vue.use(money, {precision: 4})
Vue.use(Vuelidate)

new Vue({
  router,
  render: h=>h(App),
  created(){
        
        this.$firebase.auth().setPersistence(this.$firebase.auth.Auth.Persistence.SESSION),
        this.$firebase.auth().onAuthStateChanged( user => {
        let id_cliente = localStorage.getItem('id_cliente');
        let firebase     = this.$firebase.firestore();
            if (user) {
            
            firebase.collection('sessoes_usuarios')
            .doc(id_cliente).update({status:true})
            .then(
                console.log('usuário logado com sucesso')
            ).catch((e)=>{
                console.log(e);
                console.log('não possui log de sessão')
            });

            } else {
                
                firebase.collection('sessoes_usuarios')
                .doc(id_cliente).update({status:false})
                .then(
                    console.log('usuário deslogado com sucesso')
                ).catch((e)=>{
                console.log(e);
                console.log('não possui log de sessão')
            });

            }        
        })
  }
}).$mount('#app')