Using jq to add aditional key-value pairs to json

I have this json.

{
  "type": "FeatureCollection",
  "features": [
    {
      "type": "Feature",
      "geometry": {
        "type": "MultiPolygon",
        "coordinates": [
          [
            [
              [
                4.991669,
                52.324436,
                0
              ]
            ]
          ]
        ]
      },
      "properties": {
        "neighbourhood": "Bijlmer-Oost",
        "neighbourhood_group": null
      }
    },
    {
      "type": "Feature",
      "geometry": {
        "type": "MultiPolygon",
        "coordinates": [
          [
            [
              [
                5.079162,
                52.388654,
                0
              ]
            ]
          ]
        ]
      },
      "properties": {
        "neighbourhood": null,
        "neighbourhood_group": null
      }
    }
  ]
}

What I want is to add to the properties part a new key-value pair.
Someone could help me to figure it out what is the best way to do this?

I tried with this:
jq --arg dd "$d" '.features[] |= . + {properties: download_date: $dd}' /tmp/file-$d.geojson

But this is updates the whole properites like this:
"properties": {
"download_date": "2023-03-09"
}

Expected result should look like this:

{
  "type": "FeatureCollection",
  "features": [
    {
      "type": "Feature",
      "geometry": {
        "type": "MultiPolygon",
        "coordinates": [
          [
            [
              [
                4.991669,
                52.324436,
                0
              ]
            ]
          ]
        ]
      },
      "properties": {
        "neighbourhood": "Bijlmer-Oost",
        "neighbourhood_group": null,
        "date_added": "2022-01-01"
      }
    },
    {
      "type": "Feature",
      "geometry": {
        "type": "MultiPolygon",
        "coordinates": [
          [
            [
              [
                5.079162,
                52.388654,
                0
              ]
            ]
          ]
        ]
      },
      "properties": {
        "neighbourhood": null,
        "neighbourhood_group": null,
        "date_added": "2022-01-01"
      }
    }
  ]
}

>Solution :

You can use:

$ jq --arg dd "$d" '.features[].properties += {date_added: $dd}' file