Genus bound from the s-invariant for surfaces in the tangent disk bundle of S^2

Let $DTS^2$ be the unit disk bundle of the tangent bundle of $S^2$, with boundary $\mathbb{RP}^3$. Let $(\Sigma,K)\subset(DTS^2,\mathbb{RP}^3)$ be a properly embedded orientable connected surface whose boundary is a knot $K\subset\mathbb{RP}^3$, and let $[\Sigma]^2$ denote its relative self-intersection. Genus-bound conjecture. One should have

$$2g(\Sigma)\geq-s(K)-\frac{[\Sigma]^2}{2}.$$

This conjecture extends the known genus bounds for null-homologous surfaces in $DTS^2$ and for arbitrary slice surfaces in $\mathbb{CP}^2\backslash B^4$.