Non-orientable genus of knots in punctured Spin 4-manifolds

For a closed 4-manifold $X$ and a knot $K$ in the boundary of punctured $X$, we define $\gamma_X^0(K)$ to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured $X$ with boundary $K$. Note that $\gamma^0_{S^4}$ is equal to the non-orientable 4-ball genus and hence $\gamma^0_X$ is a generalization of the non-orientable 4-ball genus. While it is very likely that for given $X$, $\gamma^0_X$ has no upper bound, it is difficult to show it. In fact, even in the case of $\gamma^0_{S^4}$, its non-boundedness was shown for the first time by Batson in 2012. In this paper, we prove that for any Spin 4-manifold $X$, $\gamma^0_X$ has no upper bound.