On a manifold with boundary $\Bar{M}$, we study a boundary value problem. We prove that, for every Yang–Mills connection $A$ with a certain nondegenracy condition and every small perturbation $\gamma$ of the boundary value $A_{|\partial \Bar{M}}$ , there is a Yang–Mills connection in the bulk that extends $A _{|\partial \Bar{M}} + \gamma$ to the bulk. We then confirm a conjecture by Cattaneo et al mentioned in their paper [15].
