Multi-select fields wit API (anonymous request)

HI! I'm trying to create a ticket via an anonymous API call that looks like this (see below). 
The tickets get created, but the one custom field that is a "multi-select" doesn't get populated with the specified values. Any ideas?

curl -X POST \
  -H "Content-Type: application/json" \
  -d '{
        "request": {
            "requester": {
                "name": "john doe",
                "email": "john.doe@example.com"
            },
            "subject": "test subject",
            "comment": {
                "body": "lorem ipsum blah blah blah",
                "public": false
            },
            "description": "test description",
            "custom_fields": [
                {
                    "id": "000000000",
                    "value": "this_comes_through_just_fine"
                },
                {
                    "id": "1111111111",
                    "value": "this_comes_through_as_well"
                },
                {
                    "id": "2222222222", // multi-select field
                    "value": [
                        "this_doesn_t_come_through",
                        "this_doesn_t_either"
                    ]
                }
            ],
            "tags": ["some_tag"]
        }
    }' \ 
https://myorg.zendesk.com/api/v2/requests

I tried to mimick the format described here, but to no avail: https://developer.zendesk.com/documentation/ticketing/managing-tickets/creating-and-updating-tickets/#setting-custom-field-values

Thanks in advance! Manu