Can I use JQL on a custom field object property

[Kiley Poole]

I am trying to write JQL that filters based on a custom field object's property but I seem to be running into issues doing so.

I have an custom field with type object that is shaped similar to this: 

"customfield_10006": [
  {
    "id": 755,
    "name": "Test",
    "startDate": "2022-08-01T19:39:00.000Z",
  }
]

I am trying to write a JQL statement that is:
cf[10004][startDate] >= 2022-08-01

But I keep getting errors addressing the id/name. Is it even possible to do JQL over custom field object properties?