Throughout: let $\otimes$ denote the minimal (i.e. spatial) $\newcommand{\Cst}{{\rm C}^*}\Cst$-tensor product of two $\Cst$-algebras.

Let $B$ be a unital, nuclear $\Cst$-algebra and let $A\subset B$ be a unital $\Cst$-subalgebra. We may identify $A\otimes B$ isometrically with a $\Cst$-subalgebra of $B\otimes B$; likewise with $B\otimes A$. Moreover, there is an isometric inclusion of algebras $A\otimes A \subseteq (A\otimes B)\cap (B\otimes A)$.

Question.Let $\sigma:B\otimes B\to B\otimes B$ denote the flip map, and let $w\in (A\otimes B)\cap (B\otimes A)$ satisfy $\sigma(w)=w$. Does it follow that $w\in A\otimes A$?

I suspect the answer is "no" but I haven't succeeded in manufacturing a counterexample. On the other hand, if the question has a positive answer, this would provide the missing piece for something I'm trying to do with group $\Cst$-algebras and related objects.

By the way: I think the answer is yes if $A$ has the operator approximation property, by the results of Kraus's 1991 JFA paper. However, in the setting I'm interested in, $A$ might not have the OAP.