Absolutely continuous laws on a bounded interval are completely mixable for all large $n$

A distribution $F$ is $n$-completely mixable if there are $n$ random variables, each with law $F$, whose sum is almost surely constant. Puccetti, Wang and Wang proved that a law on $[a,b]$ whose density is at least $3/(n(b-a))$ on all of $[a,b]$ is $n$-completely mixable, and Wang asked whether every absolutely continuous law on a bounded interval is $n$-completely mixable for all sufficiently large $n$. We answer this in the affirmative, and more generally show that every law on a bounded interval with a nonzero absolutely continuous part is $n$-completely mixable for all large $n$. The main step is quantitative: a law on an interval of length $R$ that dominates $q$ times the uniform law on an interval of length $2H$ is $n$-completely mixable for every $n\ge(6R/H+1)/q$, whatever the rest of the law is. Its proof couples the copies outside the uniform component on a quantile grid, so that their sum is within $R$ of its mean, and absorbs the error by linear tilts of the uniform copies that average out to the uniform law. The general case follows by grouping the copies in pairs, since the law of the sum of two independent copies dominates a measure with a continuous nonzero density.