Update Quote Item Gift Message

How can I update the quote item custom gift message? I’m thinking of adding that function inside the Module_Checkout/Model/Cart.php inside the function updateItems. So if I take a look at this file the content is

public function updateItems($data)
{
    $infoDataObject = new MagentoFrameworkDataObject($data);
    $this->_eventManager->dispatch(
        'checkout_cart_update_items_before',
        ['cart' => $this, 'info' => $infoDataObject]
    );

    $qtyRecalculatedFlag = false;
    foreach ($data as $itemId => $itemInfo) {
        $item = $this->getQuote()->getItemById($itemId);
        if (!$item) {
            continue;
        }

        if (!empty($itemInfo['remove']) || isset($itemInfo['qty']) && $itemInfo['qty'] == '0') {
            $this->removeItem($itemId);
            continue;
        }

        $qty = isset($itemInfo['qty']) ? (double)$itemInfo['qty'] : false;
        if ($qty > 0) {
            $item->setQty($qty);

            if ($item->getHasError()) {
                throw new MagentoFrameworkExceptionLocalizedException(__($item->getMessage()));
            }

            if (isset($itemInfo['before_suggest_qty']) && $itemInfo['before_suggest_qty'] != $qty) {
                $qtyRecalculatedFlag = true;
                $this->messageManager->addNoticeMessage(
                    __('Quantity was recalculated from %1 to %2', $itemInfo['before_suggest_qty'], $qty),
                    'quote_item' . $item->getId()
                );
            }
        }
    }

    if ($qtyRecalculatedFlag) {
        $this->messageManager->addNoticeMessage(
            __('We adjusted product quantities to fit the required increments.')
        );
    }

    $this->_eventManager->dispatch(
        'checkout_cart_update_items_after',
        ['cart' => $this, 'info' => $infoDataObject]
    );

    return $this;
}

So I want to add my function just after the condition of if ($qty > 0) { how can I save the quote item gift message?