Invariance of the image of the K3 characteristic map under the involution

Let $X$ be a 4-manifold. The involution on $\operatorname{Wh}_1(\pi_1 X;\mathbb{Z}_2\times\pi_2 X)$ is induced by the involution on $(\mathbb{Z}_2\times\pi_2 X)[\pi_1 X]$, and let $\chi(K_3\mathbb{Z}[\pi_1 X])$ denote the subgroup appearing in the target of $\Theta$. Involution-invariance conjecture.

$$\overline{\chi(K_3 \mathbb{Z}[\pi_1 X])}=\chi(K_3 \mathbb{Z}[\pi_1 X]).$$

This would make the involution descend to the quotient by $\chi(K_3\mathbb{Z}[\pi_1 X])$, allowing an involution on the target of $\Theta$ when $k_1 X\ne 0$; the source gives no resolution.