On a relation between the $\mathrm{K}$-cowaist and the $\hat{\mathsf{A}}$-cowaist

The $\mathrm{K}$-cowaist $\text{K-cw}_2 (M)$ and the $\hat{\mathsf{A}}$-cowaist $\hat{\mathrm{A}}$-$\mathrm{cw}_2 (M)$ are two interesting invariants on a manifold $M$, which are closely related to the existence of the positive scalar curvature metric on $M$. In this note, we give a detailed proof of the following inequality due to Gromov: $\text{K-cw}_2 (M) \le c \hat{\mathrm{A}}$-$\mathrm{cw}_2 (M)$, where $c$ is a dimensional constant.