Let $S$ and $T$ be positive operators on a Hilbert space $\mathcal{H}$. Suppose that $S \le T$. Since the square root function is operator monotone, it follows that $S^{1/2} \le T^{1/2}$. Does the inequality
$$S^{1/2}RS^{1/2} \le T^{1/2}RT^{1/2}$$
hold for all positive operators $R$?