MUI Autocomplete label is not updated when options are updated

Question

I have a multilingual website and an Autocomplete that lists translatable labels. Whenever the language is changed, it's container fetches its options with the same IDs, but translated labels.

The problem is, although the options in the dropdown are indeed translated, the current value, itself, is not:

How do I make the Autocomplete update its value (or trigger the onChange manually) when the options are changed?

demo