Regularity of the category of commutative Hopf algebras

Ascertain whether the category of commutative Hopf algebras over a field is regular in the sense of Barr (equivalently, whether it is Barr-exact).

Background

The authors note that commutative Hopf algebras are co-protomodular and homologically self-dual in their framework, but they do not know whether the category is regular (Barr-exact). Regularity would bring strong exactness properties crucial for homological constructions.