Universally measurable sets may all be $\mathop{\Delta}\limits_\sim {}^{1}_{2}$

We produce a forcing extension of the constructible universe $\mathbf L$ in which every universally measurable set of reals is $\mathop{\Delta}\limits_\sim {}^{1}_{2}$, partially answering question CG from David Fremlin’s problem list. The analogous result for category holds in the same model.