Cannot add webhooks in version 12 rooms

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> Failed to handle command: You must be able to set state in a room ('Change settings') in order to set up new integrations.

no matter how high I set its powerlevel. When testing in a v10 room instead, everything works as expected.

Reproduce webhook setup in a version 12 room and compare it with the working version 10 case, focusing first on the permission check for setting room state. Done means a user with sufficient powerlevel can add a webhook in version 12 rooms without the reported error.

Written by the indexing model from the issue text.