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We prove a recent conjecture of Manolescu-Willis which states that the$s$-invariant of a knot in $\mathbb{RP}^3$ (as defined by them) gives a lowerbound on its null-homologous slice genus in the unit disk bundle of $TS^2$. Wealso conjecture a lower bound in the more general case where the slice surfaceis not necessarily null-homologous, and give its proof in some special cases.

Slice genus bound in $DTS^2$ from $s$-invariant