The Lickorish-Wallace theorem tells us that any closed 3-manifold $Y$ is an integer link surgery on $S^3$, which yields an oriented cobordism between $S^3$ and $Y$. Filling out the $S^3$ by a 4-ball $B^4$, we obtain a compact 4-manifold bounding $Y$, and as a corollary we have that the 3rd oriented cobordism group $\Omega^{SO}_3=0$.

Now think of spin$^c$ analogue of this situation. We start with a spin$^c$ 3-manifold $(Y,\mathfrak{t})$. The Lickorish-Wallace theorem applied to the underlying manifold $Y$ gives a cobordism $W$ which only consists of 2-handles. To fill out the $S^3$-side of $W$ by a 4-ball, we should impose the condition that the $S^3$ is endowed with the (unique) torsion spin$^c$ structure $\mathfrak{s}_0$. Thus, a temporary version for spin$^c$ Lickorish-Wallace theorem is the existence of a spin$^c$ structure on $W$ which extends $\mathfrak{t}$ and restricts to the torsion spin$^c$ structure on $S^3$. But this cannot be true in full generality, as already seen from the case when $W$ is the product cobordism. (In this case, $\mathfrak{t}$ should be the torsion spin$^c$ structure.)

But (it seems) it is a forklore that a sufficient condition to make the theorem work is requiring that $\mathfrak{t}$ is torsion. For clarity, let me state the modified theorem again:

For a link surgery cobordism $W$ from $(S^3,\mathfrak{s}_0)$ to a closed 3-manifold $Y$ and a torsion spin$^c$ structure $\mathfrak{t}$ on $Y$, we can always find a spin$^c$ structure which restricts to $\mathfrak{s}_0$ on $S^3$ and $\mathfrak{t}$ on $Y$ (so that $(Y,\mathfrak{t})$ represent 0 in $\Omega^{Spin^c}_3$).

Question. How we can prove this theorem? This theorem appears in many places, e.g. it is a key step in defining absolute $\mathbb{Q}$-gradings in Heegaard Floer homology. But those literature do not include the proof. I guess a few obstruction theory would work, but it is mysterious to me how to exploit the torsion condition to extend the spin$^c$ structure.