  We consider a family $\mathcal F$ of maps with two branches and a common
neutral fixed point $0$ such that the order of tangency at $0$ belongs to some
interval $[\alpha_0, \alpha_1]\subset (0, 1)$. Maps in $\mathcal F$ do not
necessarily share a common Markov partition. At each step a member of $\mathcal
F$ is chosen independently with respect to the uniform distribution on
$[\alpha_0, \alpha_1]$. We show that the construction of the random tower in
Bahsoun-Bose-Ruziboev \cite{BBR} with \emph{general return time} can be carried
out for random compositions of such maps. Thus their general results are
applicable and gives upper bounds for the quenched decay of correlations of
form $n^{1-1/\alpha_0+\delta}$ for any $\delta>0$.
