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

The $\mathrm{K}$-cowaist ${\text{K-cw}}_2(M)$ and the $\widehat{\mathsf{A}}$-cowaist $\widehat{\mathrm{A}}$-${\mathrm{c}\mathrm{w}}_2(M)$ 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: ${\text{K-cw}}_2(M)\le c\widehat{\mathrm{A}}$-${\mathrm{c}\mathrm{w}}_2(M)$, where $c$ is a dimensional constant.