  We provide estimates for $s^{\rm th}$ moments of biquadratic smooth Weyl
sums, when $10\le s\le 12$, by enhancing the second author's iterative method
that delivers estimates beyond the classical convexity barrier. As a
consequence, all sufficiently large integers $n$ satisfying $n\equiv
r\pmod{16}$, with $1\le r\le 12$, can be written as a sum of $12$ biquadrates
of smooth numbers.
