The 2-knot branched-cover conjecture for odd Khovanov homology

Let $K\subset S^4$ be a smooth 2-knot. Removing a small neighborhood of a point of $K$ gives a smooth link cobordism from the empty link to the unknot $U$, and hence a map

$$\Phi\colon\mathbb{Z}\longrightarrow\Lambda^*(U)=\mathbb{Z}\{1,U\}.$$

For degree reasons, write

$$\Phi(1)=n1$$

and define $n(K)\in\mathbb{N}$ to be the absolute value of the integer $n$. Let $\widetilde{S^4}_K$ denote the branched double-cover of $S^4$, branched along $K$. The 2-knot branched-cover conjecture. For every smooth 2-knot $K\subset S^4$, the number $n(K)$ is equal to the order of the first homology of $\widetilde{S^4}_K$:

$$n(K)=\left|H_1(\widetilde{S^4}_K;\mathbb{Z})\right|.$$

The invariant $n(K)$ is intended to detect smooth 2-knots: unlike the corresponding even Khovanov invariant, which is always $1$, it can be any positive odd number. The conjectured relation with the branched double-cover remains open in the supplied text.