It is known that if $(X_n)$ and $(Y_n)$ are two $(\mathscr{F}_n)$-adapted sequences of random variables such that for each $k \geq 1$ the conditional distributions of $X_k$ and $Y_k$, given $\mathscr{F}_{k-1}$, coincide a.s., then the following is true: $\|\sum X_k\|_p \leq B_p\| \sum Y_k\|_p, 1 \leq p < \infty$, for some constant $B_p$ depending only on $p$. The aim of this paper is to show that if a sequence $(Y_n)$ is conditionally independent, then the constant $B_p$ may actually be chosen to be independent of $p$. This significantly improves all hitherto known estimates on $B_p$ and extends an earlier result of Klass on randomly stopped sums of independent random variables as well as our recent result dealing with martingale transforms of Rademacher sequences.