Smooth-supported multiplicative functions in arithmetic progressions beyond the $x^{1\mathrm{/}2}$-barrier

We show that smooth-supported multiplicative functions $f$ are well-distributed in arithmetic progressions $a_1{a}_2^{-1}(\mathrm{m}\mathrm{o}\mathrm{d}q)$ on average over moduli $q\le x^{3\mathrm{/}5-\epsilon }$ with $(q,a_1{a}_2)=1$.