Convergence rate for the hedging error of a path-dependent example

We consider a Brownian functional $F=g\bigl(\int_0^T \eta(s) dW_s\bigr)$ with $g \in L_2(\gamma)$ and a singular deterministic $\eta$. We deduce the $L_2$-convergence rate for the approximation $F^{(n)} = E F + \int_0^T \phi^{(n)}(s) dW_s$ for a class of piecewise constant predictable integrands $\phi^{(n)}$ from the fractional smoothness of $g$ quantified by Besov spaces and the rate of singularity of $\eta$.