The fiberwise Fuller trace counterexample conjecture

Let $B$ be a base space supporting a family of endomorphisms $f\colon E\to E$ over $B$. Write $R_B(f)$ and $R_B(f^2)$ for the corresponding fiberwise Reidemeister traces, and let $R_B(\Psi^{2}(f))^{C_2}$ denote the fiberwise Fuller trace. Fiberwise counterexample conjecture. There is a family of endomorphisms $f$ over some base $B$ for which

$$R_B(f)=R_B(f^2)=0,$$

but

$$R_B(\Psi^{2}(f))^{C_2}\ne 0.$$

For a single endomorphism, the Reidemeister traces for divisors of $n$ are known to be a complete obstruction, whereas the paper suggests that this may fail fiberwise; finding such an example would settle the question.