[Vuejs]-VueJS a component in a layout does not update

With mounted not firing, I solved this by exchanging events between the parent and the NavItem children. As the children are in a v-for loop, I need to set and pass the index as well as a ref in order to fire the event correctly.
Now, on route change, the emitting link gets the active style and the other children set themselves to “neutral”.

working example

SideNav parent now looks like this:

    <template>
<div class="main">

      <SideNavItem v-for="(link,index) in links" 
                  :key="link.id"
                  :path="link.path"
                  :text="link.text"
                  :index="index"
                 ref="setStyle"
                  @clicked="updateStyle"  />

</div>
</template>

<script>
import SideNavItem from "./SideNavItem.vue";

export default {
  components: {
    SideNavItem
  },
  data() {
    return {
      links: [
        { id: 1, text: "GoTo Page1", path: "/" },
        { id: 2, text: "GoTo Page2", path: "/page2" },
        { id: 3, text: "GoTo Page3", path: "/page3" }
      ]
    };
  },
  methods: {
    updateStyle( index) { 

        var i;
        for (i = 0; i < this.links.length; i++) { 
           this.$refs.setStyle[i].changeStyle( index);
        }

      },
  }
};
</script>

And the child NavItem:

    <template>

  <div class="link1" @click="changePage" :style="{ backgroundColor: 
  setBackground, color:setColor }">
  {{ text }}
  </div>

</template>
<script>
export default {
  props: {
    path: { type: String },
    text: { type: String },
    index: { type: Number},
  },
  data() {
    return {
      myBackgroundColor: "",
      myColor: ""
    };
  },
  computed: {
    setBackground() {
      return this.myBackgroundColor;
    },
    setColor() {
      return this.myColor;
    }
  },
  methods: {
       changePage(){

            this.$emit('clicked',this.index)

            this.$router.push({ path: this.path })
        }, 
        changeStyle(index) {

            if (this.index === index) {
            this.myBackgroundColor = "blue";
            this.myColor = "white";
            }
            else  {
             this.myBackgroundColor = "";
            this.myColor = "";
            }

        },
  },

};
</script>