  By considering negative surgeries on a knot $K$ in $S^3$, we derive a lower
bound to the non-orientable slice genus $\gamma_4(K)$ in terms of the signature
$\sigma(K)$ and the concordance invariants $V_i(\overline{K})$, which
strengthens a previous bound given by Batson, and which coincides with
Ozsv\'ath-Stipsicz-Szab\'o's bound in terms of their $\upsilon$ invariant for
L-space knots and quasi-alternating knots. A curious feature of our bound is
superadditivity, implying, for instance, that the bound on the stable
non-orientable genus is sometimes better than the one on $\gamma_4(K)$.
