Thank you for the response!
Indeed, that discussion (and the traces to here and here) were immensely useful. The latter link summarizes best why covariance structure is necessary and why the posterior still shows the covariance. A lot to wra my head around.
I will follw the useful sleep study notebook by @Jack_Caster and your notebook. If I manage to adapt something for my example, I will post it here.
Best,
Falk