Let $A, B$ be $N \times N$ symmetric positive definite matrices and $A^{1/2}, B^{1/2}$ their symmetric, positive definite square roots. Furthermore, suppose that $$ C = A-B$$ has rank at most $R \in \mathbb N$, with $ R \ll N$.
Is it true, that the rank of
$$ D = A^{1/2} - B^{1/2}$$ may be bounded by a constant multiple of $R$?