Xu's non-vanishing conjecture for connected sums with a four-torus

Let $X_i$ for $i=1,2,3$ be either a 4-torus $T^4$, or a closed oriented almost complex 4-manifold with

$$b_1(X_i)=0,$$ $$b^+(X_i)\equiv 3\pmod 4$$

and $SW_{X_i}(\Gamma_{X_i})\equiv 1\pmod 2$, where $\Gamma_{X_i}$ is a spin$^c$ structure compatible with the almost complex structure. Xu's conjecture. The connected sum

$$\left(\#^{\ell}_{i=1}X_i\right)\# T^4$$

has a non-trivial stable cohomotopy Seiberg–Witten invariant for $\ell=2,3$. The conjecture extends known non-vanishing results for connected sums with simply connected summands and with a four-torus; the source states that it remains open.