Deletable Chips in Material UI Multiple Select

The Material UI documentation includes an example of a multiple select where the selected options are rendered with the Chip component by using the renderValue prop on the Select. The standard behavior for the Select component is that clicking on the current value opens the list of options.

I am trying to tweak this so that the Chips show the X button, and clicking on the X should instantly remove that item from the selections rather than opening the options list.

That seems easy, but I can't get the onDelete event of the Chip to fire. Clicking the X still just opens the Select.

How can I get the onDelete event to take priority? From what I know about event bubbling, it seems like the Chip should handle the event first.

Code Sandbox Demo

Code:

const MultipleSelectDemo = () => {
  const [personName, setPersonName] = React.useState<string[]>(initialSelected);

  const handleChange = (event: React.ChangeEvent<{ value: unknown }>) => {
    setPersonName(event.target.value as string[]);
  };
  
  // this never gets called
  const handleDelete = (e: React.MouseEvent, value: string) => {
    e.preventDefault();
    console.log("clicked delete");
    setPersonName((current) => _without(current, value));
  };

  return (
      <div>
        <FormControl>
          <InputLabel id="demo-mutiple-chip-checkbox-label">
            Chip + Check
          </InputLabel>
          <Select
            labelId="demo-mutiple-chip-checkbox-label"
            id="demo-mutiple-chip-checkbox"
            multiple
            value={personName}
            onChange={handleChange}
            onOpen={() => console.log("select opened")}
            IconComponent={KeyboardArrowDownIcon}
            renderValue={(selected) => (
              <div>
                {(selected as string[]).map((value) => (
                  <Chip
                    key={value}
                    label={value}
                    clickable
                    className={classes.chip}
                    onDelete={(e) => handleDelete(e, value)}
                    onClick={() => console.log("clicked chip")}
                  />
                ))}
              </div>
            )}
          >
            {names.map((name) => (
              <MenuItem key={name} value={name}>
                <Checkbox checked={personName.includes(name)} />
                <ListItemText primary={name} />
              </MenuItem>
            ))}
          </Select>
        </FormControl>
      </div>
  );
}