The Gapeev-Shiryaev conjecture (originating in Gapeev and Shiryaev (2011) and Gapeev and Shiryaev (2013)) can be broadly stated as follows: Monotonicity of the signal-to-noise ratio implies monotonicity of the optimal stopping boundaries. The conjecture was originally formulated both within (i) sequential testing problems for diffusion processes (where one needs to decide which of the two drifts is being indirectly observed) and (ii) quickest detection problems for diffusion processes (where one needs to detect when the initial drift changes to a new drift). In this paper we present proofs of the Gapeev-Shiryaev conjecture both in (i) the sequential testing setting (under Lipschitz/Holder coefficients of the underlying SDEs) and (ii) the quickest detection setting (under analytic coefficients of the underlying SDEs). The method of proof in the sequential testing setting relies upon a stochastic time change and pathwise comparison arguments. Both arguments break down in the quickest detection setting and get replaced by arguments arising from a stochastic maximum principle for hypoelliptic equations (satisfying Hormander's condition) that is of independent interest. Verification of the Gapeev-Shiryaev conjecture establishes the fact that sequential testing and quickest detection problems with monotone signal-to-noise ratios are amenable to known methods of solution.