ANGULAR - How to keep select dropdown opened when opening mat-menu?

I have a mat-select component that contains custom options (with additional menu "Set as default"):

<mat-form-field appearance="fill">
  <mat-label>Favorite food</mat-label>
  <mat-select>
    <mat-option *ngFor="let food of foods" [value]="food.value">
      {{food.viewValue}}
      <button mat-icon-button [matMenuTriggerFor]="menu" aria-label="Example icon-button with a menu">
        <mat-icon>more_vert</mat-icon>
      </button>
      <mat-menu #menu="matMenu">
        <button mat-menu-item>
          <span>Set as default</span>
        </button>
      </mat-menu>
    </mat-option>
  </mat-select>
</mat-form-field>

However, when I open the menu, select option list disappears and I can see only "Set as default" pop-up.

Can we show "Set as default" on top of the select option list somehow or it is impossible in Angular Material?

This is a link to Stackblitz with the code: https://stackblitz.com/edit/angular-material-playground-hcw1wo?file=src/app/app.component.html