On a conjecture of A. Bikchentaev

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 $b^{1\mathrm{/}2}a{b}^{1\mathrm{/}2}$ 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.