  We prove that the condition \begin{equation}
\sum_{n=1}^\infty\frac{1}{nw(n)}<\infty \end{equation} is necessary for an
increasing sequence of numbers $w(n)$ to be an almost everywhere unconditional
convergence Weyl multiplier for the trigonometric system. This property for
Haar, Walsh, Franklin and some other classical orthogonal systems was known
long ago. The proof of this result is based on a new sharp logarithmic lower
bound on $L^2$ of the majorant operator related to the rearranged trigonometric
system.
