Add Shipping Method Description to estimate-shipping-methods-by-address-id response

I am trying to show additional information about the Shipping Method in Checkout. I could already save this information in the column method_description in quote_shipping_rate. And with a plugin, I add the data as an extension attribute:

extension_attributes.xml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Api/etc/extension_attributes.xsd">
  <extension_attributes for="MagentoQuoteApiDataShippingMethodInterface">
    <attribute code="method_description" type="string" />
  </extension_attributes>
</config>

di.xml

<?xml version="1.0" encoding="utf-8"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
  <type name="MagentoQuoteModelCartShippingMethodConverter">
    <plugin name="vendor_carrier_plugin_shipping_method_converter" type="VendorCarrierPluginCartShippingMethodConverterPlugin" sortOrder="10" disabled="false" />
  </type>
</config>

ShippingMethodConverter.php

<?php
namespace VendorCarrierPluginCart;

class ShippingMethodConverterPlugin
{
  /**
   * @param MagentoQuoteModelCartShippingMethodConverter $subject
   * @param MagentoQuoteApiDataShippingMethodInterface $result
   * @param MagentoQuoteModelQuoteAddressRate $rateModel
   * @param string $quoteCurrencyCode
   * @return MagentoQuoteApiDataShippingMethodInterface
   */
  public function afterModelToDataObject(
    MagentoQuoteModelCartShippingMethodConverter $subject,
    MagentoQuoteApiDataShippingMethodInterface $result,
    MagentoQuoteModelQuoteAddressRate $rateModel,
    string $quoteCurrencyCode
  ){
    if($rateModel->getMethodDescription()){
      $extensionAttributes = $result->getExtensionAttributes();
      $extensionAttributes->setData('method_description', $rateModel->getMethodDescription());
      $result->setExtensionAttributes($extensionAttributes);
    }

    return $result;
  }
}
?>

I tested it, and it looks to work. If in the class MagentoQuoteModelShippingMethodManagement in the method getShippingMethods, I logged the content this just before return $output:

foreach($output as $_output){
    $logger->info(print_r($_output->getExtensionAttributes()->__toArray(), true));
}

It returns me the correct data:

Array
(
    [method_description] => Delivery Date: 2022-11-17T23:59:00
)

However, in checkout the response of the rest/default/V1/carts/mine/estimate-shipping-methods-by-address-id doesn’t have this data in the JSON:

[ {
“carrier_code”: “freeshipping”,
“method_code”: “freeshipping”,
“carrier_title”: “Free Shipping”,
“method_title”: “Free”,
“amount”: 0,
“base_amount”: 0,
“available”: true,
“error_message”: “”,
“price_excl_tax”: 0,
“price_incl_tax”: 0 } ]

What could I be missing to add the method_description field in this response?