Miao's majorisation conjecture for concave singular-value functions

Let $X,Y\in\mathbb{M}_{n,m}(\mathbb{C})$, and let $\sigma_i(X)$ denote the singular values in non-increasing order. Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be concave with $f(0)=0$. Miao's conjecture. For every $k\leq n,m$,

$$\sum_{i=1}^k\left|f(\sigma_i(X))-f(\sigma_i(Y))\right|\leq\sum_{i=1}^k f(\sigma_i(X-Y)).$$

This strengthens the one-sided majorisation inequality stated immediately before it; the source attributes the conjecture to W. Miao and gives no resolution.