  We show under $\sf{ZF} + \sf{DC} + \sf{AD}_{\mathbb{R}}$ that every set of
reals is $I$-regular for any $\sigma$-ideal $I$ on the Baire space
$\omega^{\omega}$ such that $\mathbb{P}_I$ is proper. This answers the question
of Khomskii. We also show that the same conclusion holds under $\sf{ZF} +
\sf{DC} + \sf{AD}^+$ if we additionally assume that the set of Borel codes for
$I$-positive sets is $\mathbf{\Delta}^2_1$. If we do not assume $\sf{DC}$, the
notion of properness becomes obscure as pointed out by Asper\'{o} and Karagila.
Using the notion of strong properness similar to the one introduced by Bagaria
and Bosch, we show under $\sf{ZF} + \sf{DC}_{\mathbb{R}}$ without using
$\sf{DC}$ that every set of reals is $I$-regular for any $\sigma$-ideal $I$ on
the Baire space $\omega^{\omega}$ such that $\mathbb{P}_I$ is strongly proper
assuming every set of reals is $\infty$-Borel and there is no
$\omega_1$-sequence of distinct reals. In particular, the same conclusion holds
in a Solovay model.
