with Neumann boundary conditions). To conclude the convergence, it remains to prove the uniqueness of energy solutions. We achieve this following the same strategy used in [@Gubinelli2015Energy] for proving the uniqueness of energy solutions on the real line: We mollify an energy solution ${{\mathcal Y}}$ to the SBE, ${{\mathcal Y}}^n$, find a suitable anti-derivative ${{\mathcal Z}}^n$ of ${{\mathcal Y}}^n$, and let $\Phi^n = e^{{{\mathcal Z}}^n}$. Now we take the mollification away and show that $\Phi = \lim_n \Phi^n$ solves (a version of) the SHE. Since uniqueness holds for solutions to the SHE, ${{\mathcal Y}} = \nabla \log \Phi$ must be unique. But while the strategy is the same as in [@Gubinelli2015Energy], the technical details are considerably more involved because our space is no longer translation invariant and many of the tools of [@Gubinelli2015Energy] break down. Moreover, the dynamics of $\Phi^n$ contain a singular term that converges to Dirac deltas at the boundaries, a new effect which can be interpreted as a change of boundary conditions: Formally we would expect that $\nabla \Phi_t(0) = e^{{{\mathcal Z}}_t(0)} \nabla {{\mathcal Z}}_t(0) = 0$ because $\nabla {{\mathcal Z}}_t(0) = 0$ by the Neumann boundary conditions for ${{\mathcal Z}}$. However, the singular term in the 