Vanishing conjecture for PU(2) monopole link intersections

Let $X$ be a closed, oriented, smooth four-manifold with $b_2^+(X)>0$. Let ${\mathfrak{s}}={\mathfrak{s}}_0\otimes L_1$ be a $\operatorname{Spin}^c$ structure defined by ${\mathfrak{s}}_0$ and a reduction

$$E_{-\ell}=L_1\oplus (\det E)\otimes L_1^*,$$

where $c_2(E_{-\ell})=c_2(E)-\ell$ and $\ell\geq 0$. Let ${\mathbf{L}}_{W,E,L_1}$ be the link of the ideal Seiberg–Witten moduli space $M^{sw}_{\mathfrak{s}}\times \operatorname{Sym}^{\ell}(X)$. Vanishing conjecture. The intersection number

$$\#\left({\bar{V}}(z)\cap {\bar{W}}(x^{n_a-1})\cap {\mathbf{L}}_{W,E,L_1}\right)$$

appearing in the right-hand side of Equation~ is a multiple of $SW_X({\mathfrak{s}})$ and therefore vanishes when $SW_X({\mathfrak{s}})=0$. This conjecture would reduce the sum over reducible $\operatorname{PU}(2)$ monopoles to contributions from $\operatorname{Spin}^c$ structures with nonzero Seiberg–Witten invariant; the supplied passage gives no resolution status.