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We prove the conjecture for the sphere. Namely when $D$, the diameter of a convex domain in the unit $S^n$ sphere, is $\\le \\frac{\\pi}{2}$, the gap is greater than the gap of the corresponding $1$-dim sphere model. We also prove the gap is $\\ge 3\\frac{\\pi^2}{D^2}$ when $n \\ge 3$, giving a sharp bound. 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