[en] The so-called negative mode problem in the path integral approach to the false vacuum decay with the effect of gravity has been an unsolved problem. Several years ago, we proposed a conjecture which is to be proved in order to give a consistent solution to the negative mode problem. We called it the 'no-negative mode conjecture'. In the present paper, we give a proof of this conjecture for rather general models. Recently, we also proposed the 'no-supercritical supercurvature mode conjecture' that claims the absence of supercritical supercurvature modes in the one-bubble open inflation model. In the same paper, we clarified the equivalence between the 'no-negative mode conjecture' and the 'no-supercritical supercurvature mode conjecture'. Hence, the latter is also proved at the same time when the former is proved