For two vectors $x$ and $y$ in $\mathbb{R}^n$, recall that $y$ weakly majorizes $x$, denoted by $x\prec_w y$, if the sum of the $k$ largest entries of $x$ is smaller than or equal to that of $y$ for each $k \in\{1, ..., n\}$.
For any matrix $A$, we use $\sigma(A)$ to denote the vector consisting of the singular values of $A$ in descending order.
Is the following "theorem" true if $A$ and $B$ are not square?
Theorem. For any matrices $A$ and $B$ of the same size, we have $|\sigma(A)-\sigma(B)| \prec_w \sigma(A-B)$, where the absolute value is taken entrywise.
This theorem is indeed true when $A$ and $B$ are both square matrices, and it is Theorem 2.3 of [A. S. Markus, The eigen- and singular values of the sum and product of linear operators, Uspekhi Mat. Nauk, 1964, Volume 19, Issue 4(118), 93–123]. [Markus 1964] attributes this result to [L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. II (1960), 50-59], which mentions it in the proof of Theorem 5. Theorem 5.4 of [Markus 1964] extends it to completely continuous operators on a separable Hilbert space. The inequality is also given in (2.9) of [T. Ando, Majorizations and inequalities in matrix theory. Linear algebra and its Applications, 1994, vol. 199, p. 17-67].
Question. Does the abovementioned theorem still hold when $A$ and $B$ are not square? It seems yes to me, and the proof in [Markus 1964] is still applicable. However, I have not found a reference stating this theorem in the non-square case and providing a proof. It would be great if anyone can direct me to such a reference. Thank you very much.
P.S.: A "simple" consequence of this "theorem" would be, for example, the Hoffman-Wielandt bound for singular values: $\|\sigma(A) - \sigma(B)\|_2 \le \|A-B\|_\text{F}$, which does hold in the rectangular case (e.g., [Corollary 7.3.5, R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, second edition, 2012]).
Update: Reference found. See Theorem 3.4.5 in [R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1994] and Theorem 10.24 in [Zhang, Matrix Theory: Basic Results and Techniques, Springer, New York, second edition, 2011]. Many thanks to @ToniMhax.