Context.
The Drinfeld center of a rigid monoidal category $\mathcal{C}$ is again rigid. This is not hard to see: Given an object $X\in \mathcal{C}$ together with a half-braiding $\phi_X:X\otimes-\xrightarrow{\sim} -\otimes X$, one can endow the left (resp. right) dual object of $X$ with a half-braiding using the respective evaluations and coevaluations in $\mathcal{C}$. For details see Theorem 2.23 in this PhD thesis, for instance.

I am interested in the Drinfeld center of non-rigid closed monoidal categories. For $R$ a commutative ring, the Drinfeld center of the symmetric monoidal category $\operatorname{Mod}(R)$ of (left) $R$-modules is monoidally equivalent to $\operatorname{Mod}(R)$ itself; see here. In particular, it is closed monoidal.

Question.
Is the Drinfeld center of a closed monoidal category always closed monoidal? Is the Drinfeld center of a (non-symmetric) $\ast$-autonomous category again $\ast$-autonomous? Have Drinfeld centers of non-rigid closed monoidal categories been considered in the literature? If so, where?

Ideally, someone has described the Drinfeld centers of a few naturally ocurring non-rigid closed monoidal categories by showing that they are each monoidally equivalent to another known monoidal category. Such a stock of examples might be useful to me in testing some ideas and conjectures.