A Ky Fan singular-value inequality for concave functions

Let $A,B\in\mathbb{R}^{m\times n}$ be given, and let $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ be a concave function satisfying $f(0)=0$. For $k\in\{1,\ldots,\min\{m,n\}\}$, let $\sigma_i(A)$ and $\sigma_i(B)$ denote the singular values of $A$ and $B$, ordered nonincreasingly. The concave singular-value conjecture. For every such $k$,

$$\sum_{i=1}^k\bigl|f(\sigma_i(A))-f(\sigma_i(B))\bigr|\leq\sum_{i=1}^k f(\sigma_i(A-B)).$$

This conjecture proposes a generalization of a perturbation inequality for singular values and would provide further applications to low-rank matrix recovery. Its resolution is not given in the source.