This question is related to one I asked here in Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM. In that question I asked if anyone knew of an example of a Cohen Macaulay, $ \mathbb{Q} $-factorial, normal, Mori dream space $ Z $ over a field $ k $ of arbitrary characteristic such that the Cox ring of $ Z $ is integral, but not Cohen Macaulay.
Jason Starr gave an answer that every Abelian variety $ Z $ of very general type such that $ \dim(Z) \ge 2 $ satisfies this property. In my case it is very relevant that the underlying variety $ Z $ is ruled, i.e., it is birational to a variety of the form $ Y \times \mathbb{P}^{1}_{k} $. Does anyone know of a $ \mathbb{Q} $-factorial, ruled, Cohen Macaulay, projective, Mori dream space $ Z $ such that the Cox ring of $ Z $ is integral and not Cohen Macaulay?
