Mod-16 conjecture for the torus Furuta–Ohta invariant

Let $X$ be an integral homology $S^1 \times S^3$ containing an essentially embedded torus $\mathcal{T}$, let $Y_0$ be the cross-section obtained by the associated $0$-surgery, and suppose there is an oriented $4$-manifold $M$ with $\partial M=Y_0$ such that the representation $\varphi_{2\alpha}$ extends over $\pi_1(M)$. Let $\sigma_{\varphi_{2\alpha}}(M)$ be the twisted signature and let $\alpha\in(0,1/2)$ be a non-degenerate holonomy parameter. Mod-16 conjecture.

$$\lambda_{FO}(X,\mathcal{T},\alpha)+\sigma_{\varphi_{2\alpha}}(M)=0\mod 16.$$

This conjecture is motivated by Furuta–Ohta's mod-2 conjecture and the preceding relation with the signature invariant; the supplied text gives no resolution.