Equality of the two pseudo-isotopy obstruction subgroups

Let $X$ be a 4-manifold with $k_1 X=0$. The subgroups $\Theta(\mathcal{J}(X))$ and $\Theta(\mathcal{J}(X)\cap\ker\Sigma)$ are defined in the target of the pseudo-isotopy invariant $\Theta$. Equality conjecture.

$$\Theta(\mathcal{J}(X))=\Theta(\mathcal{J}(X)\cap\ker\Sigma).$$

If true, the quotient used to define $\Theta$ when $\Sigma(f)\ne 0$ would agree with the quotient obtained by restricting to $\mathcal{J}(X)\cap\ker\Sigma$; the source provides no resolution of this claim.