Enumerative P=W conjecture for the defect-homogeneous extension

Let $Z$ be the cohomological space under consideration, and let $\operatorname{def}$ denote its defect grading. For integers $k\geq 1$ and $0\leq h\leq k$, let the redundancy range be the range of genera satisfying $3g\geq 3k+h+1$. Let $F\in H^*(Z)$ and let $P\in H^*(Z)$ be defect-homogeneous classes, and let Equation

denote the extension equation from the source. **Enumerative P=W conjecture.** In the redundancy range, there \exists a unique defect-homogeneous $F\in H^*(Z)$ with $\operatorname{def}(F)=2k-2$ such that, for every defect-homogeneous $P\in H^*(Z)$ with $\operatorname{def}(P)=2k-2$, Equation

is satisfied.

The claim is the top-defect part of the enumerative $P=W$ conjecture. The case $h=0$ was proved in the preceding section; the displayed assertion concerns the general case in the redundancy range.