Let $H$ be a Hopf algebra with invertible antipode. Let $A$ be a braided Hopf algebra in the Yetter-Drinfeld category ${}_H^H\mathcal{YD}$ over $H$.

A nonzero left integral in $A$ is a nonzero element $x\in A$ such that $yx=\epsilon(y)x$ for all $y\in A$. A nonzero right integral in $A$ is a nonzero element $x\in A$ such that $xy=\epsilon(y)x$ for all $y\in A$.

If $A$ is a classical Hopf algebra with trivial braiding, it is known that the existence of a nonzero left integral or nonzero right integral in $A$ implies finite dimensionality of $A$. This is a result due to Sweedler, Integrals for Hopf algebras, Annals of Mathematics, 1969.

My question concerns the braided case: Is the analogous result true for every braided Hopf algebra $A$? Does the existence of a nonzero left integral or nonzero right integral in $A$ imply finite dimensionality?

A reference for this general result would be welcome. If it helps, you could assume additionally that $A$ is a $\mathbb{Z}_{\geq 0}$-graded braided Hopf algebra, and if it helps further, you could assume additionally that $A$ is a connected $\mathbb{Z}_{\geq 0}$-graded braided Hopf algebra (i.e. $A^0=\mathbf{k}1$ where $\mathbf{k}$ is the ground field).

**Remark.** The proof of Sweedler for classical Hopf algebras does not directly carry over to the braided case.