  The nonorientable 4-genus $\gamma_4(K)$ of a knot $K$ is the smallest first
Betti number of any nonorientable surface properly embedded in the 4-ball, and
bounding the knot $K$. We study a conjecture proposed by Batson about the value
of $\gamma_4$ for torus knots, which can be seen as a nonorientable analogue of
Milnor's Conjecture for the orientable 4-genus of torus knots. We prove the
conjecture for many infinite families of torus knots, by relying on a lower
bound for $\gamma_4$ formulated by Ozsv\'ath, Stipsicz, and Szab\'o. As a side
product we obtain new closed formulas for the signature of torus knots.
