Products of primes in arithmetic progressions: a footnote in parity breaking

We prove that, if $x$ and $q⩽x^{1\mathrm{/}16}$ are two parameters, then for any invertible residue class $a$ modulo $q$ there exists a product of exactly three primes, each one below $x^{1\mathrm{/}3}$, that is congruent to $a$ modulo $q$.