How I can use Variables in Ansible json_query?

I want to use a ansible variable inside the json query filter. This is my Code:

Playbook execution:

ansible-playbook debug.yml -e "project_environment=live"

- debug:
    msg: "{{ project_environment }}"

- debug:
    msg: "{{ check_objects | json_query('`{{project_environment}}`.current') }}"

This is my dictionary:

check_objects:
  live:
    current:
    - example.com
    next:
    - next.example.com

This is what i got:

TASK [debug : debug] 
ok: [sample-hostname] => {
    "msg": "live"
}

TASK [debug  : debug]
ok: [sample-hostname] => {
    "msg": ""
}

When i replace the variable by using the expected value, the output it is working fine:

- debug:
    msg: "{{ check_objects | json_query('live.current') }}"

TASK [typo3-deployment/check : debug] 
ok: [sample-hostname] => {
    "msg": [
        "example.com"
    ]
}

I think it runs in trouble while interpolate the variable.

I have tried this solution but it doesn't work too: Ansible : pass a variable in a json_query filter

The task with json_query below

  vars:
    project_environment: live
  tasks:
    - debug:
        msg: "{{ check_objects|
                 dict2items|
                 json_query(query)|
                 flatten }}"
      vars:
        query: "[?key=='{{ project_environment }}'].value.current"

gives

"msg": [
    "example.com"
]

The same result can be achieved also with the task

- debug:
    var: check_objects[project_environment].current