Status bar showing idle status for a busy kernel when switching notebooks

## Description

Jupyter lab status bar is showing that the kernel is idle even though the kernel is busy as seen in the top right kernel status tab. This occurs when we switch between an idle notebook and a busy notebook.

## Reproduce

1. Create a new notebook named **notebook_1.ipynb**. Add a long running script in a cell and execute the same.
2. Create a second notebook named **notebook_2.ipynb**. Assign a kernel and wait until kernel is idle.
3. Switch between notebook_1.ipynb and notebook_2.ipynb.
4. Check status bar of notebook_1.ipynb. Kernel status will be shown as idle instead of busy.

## Expected behaviour

Kernel status of busy notebook should be shown as _busy_ instead of _idle_.

## Context

- Operating System and version: macOS Monterey, version 12.6.4
- Browser and version: Chrome 113.0.5672.92
- JupyterLab version: 3.6.3 and 4.0.0
- ipykernel version : 6.16.2
- ipython version : 7.34.0

package versions

aiofiles==22.1.0

aiosqlite==0.19.0

anyio==3.6.2

appnope==0.1.3

argon2-cffi==21.3.0

argon2-cffi-bindings==21.2.0

arrow==1.2.3

attrs==23.1.0

Babel==2.12.1

backcall==0.2.0

beautifulsoup4==4.12.2

bleach==6.0.0

cached-property==1.5.2

certifi==2023.5.7

cffi==1.15.1

charset-normalizer==3.1.0

debugpy==1.6.7

decorator==5.1.1

defusedxml==0.7.1

entrypoints==0.4

fastjsonschema==2.17.1

fqdn==1.5.1

idna==3.4

importlib-metadata==6.6.0

importlib-resources==5.12.0

ipykernel==6.16.2

ipython==7.34.0

ipython-genutils==0.2.0

isoduration==20.11.0

jedi==0.18.2

Jinja2==3.1.2

json5==0.9.14

jsonpointer==2.3

jsonschema==4.17.3

jupyter-events==0.6.3

jupyter-server==1.24.0

jupyter-ydoc==0.2.4

jupyter_client==7.4.9

jupyter_core==4.12.0

jupyter_server_fileid==0.9.0

jupyter_server_ydoc==0.8.0

jupyterlab==3.6.3

jupyterlab-pygments==0.2.2

jupyterlab_server==2.22.1

MarkupSafe==2.1.2

matplotlib-inline==0.1.6

mistune==2.0.5

nbclassic==1.0.0

nbclient==0.7.4

nbconvert==7.4.0

nbformat==5.8.0

nest-asyncio==1.5.6

notebook==6.5.4

notebook_shim==0.2.3

packaging==23.1

pandocfilters==1.5.0

parso==0.8.3

pexpect==4.8.0

pickleshare==0.7.5

pkgutil_resolve_name==1.3.10

prometheus-client==0.16.0

prompt-toolkit==3.0.38

psutil==5.9.5

ptyprocess==0.7.0

pycparser==2.21

Pygments==2.15.1

pyrsistent==0.19.3

python-dateutil==2.8.2

python-json-logger==2.0.7

pytz==2023.3

PyYAML==6.0

pyzmq==25.0.2

requests==2.30.0

rfc3339-validator==0.1.4

rfc3986-validator==0.1.1

Send2Trash==1.8.2

six==1.16.0

sniffio==1.3.0

soupsieve==2.4.1

terminado==0.17.1

tinycss2==1.2.1

tomli==2.0.1

tornado==6.2

traitlets==5.9.0

typing_extensions==4.5.0

uri-template==1.2.0

urllib3==2.0.2

wcwidth==0.2.6

webcolors==1.13

webencodings==0.5.1

websocket-client==1.5.2

y-py==0.5.9

ypy-websocket==0.8.2

zipp==3.15.0

https://github.com/jupyterlab/jupyterlab/assets/6435143/46594eec-75e3-4fcb-9ece-4b25360dedbd

Original issue was opened as [jupyterlab/jupyterlab#14578](https://github.com/jupyterlab/jupyterlab/issues/14578). Migration requested by [brijsiyag](https://github.com/brijsiyag)
IOpub messages received twice with alternate execution_state as `idle` and `busy`. This leads to the inconsistent behaviour.