Scalar and Mean Curvature Comparison on Compact Cylinder

Let $X$ be a closed, oriented Riemannian manifold. Denote by $(M = X \times I, \partial M = X \times \lbrace 0 \rbrace \cup X \times \lbrace 1 \rbrace, g)$ a compact cylinder with smooth boundary, $\dim M \geqslant 3$. In this article, we address the following question: If $g$ is a Riemannian metric having (i) positive scalar curvature (PSC metric) on $M$ and nonnegative mean curvature on $\partial M$; and (ii) the $g$-angle between normal vector field $\nu_{g}$ along $\partial M$ and $\partial_{\xi} \in \Gamma(TI)$ being less than $\frac{\pi}{4}$, then there exists a metric $\tilde{g}$ on $M$ such that $\tilde{g} |_{X \times \lbrace 0 \rbrace}$ is a PSC metric on $X \cong X \times \lbrace 0 \rbrace$. Equivalently, we show that if $X$ admits no PSC metric, but $M$ admits a PSC metric $g$ satisfying the angle condition, then the mean curvature on $\partial M$ must be negative somewhere. This generalizes a result of Gromov and Lawson (Ann. of Math. (2), 1980) for $X = \mathbb{T}^{n}$.