ct.category theory – Can a braided fusion category have an order-2 Morita equivalence which has no representative that is both connected and isomorphic to its opposite?

Let $mathcal{B}$ be a braided fusion category over $mathbb{C}$. Let me write $mathrm{Alg}(mathcal{B})$ for the set of isomorphism classes of unital associative algebra objects in $mathcal{B}$, and $mathrm{Mor}(mathcal{B}) = mathrm{Alg}(mathcal{B})/sim_M$ for the set of Morita equivalence classes. These sets are naturally associative (but usually noncommutative) monoids under the tensor product operation. (The formula for multiplication in a tensor product of algebra objects explicitly references the braiding in $mathcal{B}$.)

An algebra $X in mathrm{Alg}(mathcal{B})$ is called connected if the unit object $1inmathcal{B}$ appears with multiplicity $1$ in the direct sum decomposition of (the underlying object of) $mathcal{B}$. Every simple separable algebra is Morita equivalent to a connected one: given a separable algebra $X in mathrm{Alg}(mathcal{B})$, you choose a simple $X$-module, and consider its algebra of endomorphisms.

Every algebra object $X in mathrm{Alg}(mathcal{B})$ has an opposite algebra $X^{mathrm{op}}$ — the formula for multiplication in $X^{mathrm{op}}$ uses the braiding in $mathcal{B}$. An algebra $X$ is called Azumaya if its Morita equivalence class $(X) in mathrm{Mor}(mathcal{B})$ is invertible. If $X$ is Azumaya, then $(X)^{-1}$ is the Morita-equivalence class of $X^mathrm{op}$.

I am interested in Azumaya algebras which are Morita self-dual in the sense that $(X) = (X)^{-1} in mathrm{Mor}(mathcal{B})$, or in other words there exists a Morita equivalence $X sim_M X^{mathrm{op}}$. One way this can occur, of course, is if $X$ is isomorphic as an algebra to $X^{mathrm{op}}$. Let’s call such an algebra antiautomorphic. Every Morita self-dual algebra is Morita equivalent to an antiautomorphic one: what you do is to look at the category of $X$-modules; to take a direct sum of all simple $X$-modules, with each isomorphism class appearing with multiplicity one; and take the endomorphism algebra of this sum.

This universal representative, however, will typically fail to be connected. On the other hand, I don’t know very many examples, and in the examples I do know, I can always find an antiautomorphic connected representative of each self-dual Azumaya class.

What is an example of a braided fusion category $mathcal{B}$ and an order-2 element in $mathrm{Mor}(mathcal{B})$ which has no representative that is simultaneously connected and antiautomorphic?