  In \cite{bik1}, A. M. Bikchentaev conjectured that for positive
$\tau-$measurable operators $a$ and $b$ affiliated with an arbitrary semifinite
von Neumann algebra $\mathcal M$, the operator $b^{1/2}ab^{1/2}$ is
submajorized by the operator $ab$ in the sense of Hardy-Littlewood. We prove
this conjecture in full generality and present a number of applications to
fully symmetric operator ideals, Golden-Thompson inequality and (singular)
traces.
