Comparison of weak and strong moments for vectors with independent coordinates

We show that for $p\ge 1$, the $p$-th moment of suprema of linear combinations of independent centered random variables are comparable with the sum of the first moment and the weak $p$-th moment provided that $2q$-th and $q$-th integral moments of these variables are comparable for all $q \ge 2$. The latest condition turns out to be necessary in the i.i.d. case.