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Let $R^\\gamma(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-\\gamma\\Delta_g+\\mathrm{R}_g$ is strictly positive.\n  We prove that if $n=2$ and $\\gamma\\ge0$, or if $n\\ge3$ and $0\\leq \\gamma \\leq 4(n-1)/(n-2)$, the inclusion $R^0(M)\\hookrightarrow R^\\gamma(M)$ is a homotopy equivalence, thus generalizing, to all dimensions and in the maximal range, the results of Botvinnik--Rosenberg and Li--Mantoulidis. 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