Cuspidal $u$-plane formula for generalized Donaldson invariants

Let $c=\operatorname{H}$, let $N_f=0,2,3$, and let $m,n,k$ satisfy

$$2m+2n+4=(4-N_f)k.$$

Write $\pmb{\mathrm{D}}^{N_f}_{m,2n}$ for the Donaldson invariant and $\pmb{\Phi}_{k,m,2n}$ and $\pmb{\Phi}^{N_f,c,1}_{k,m,2n}$ for the corresponding $u$-plane invariants. The Moore–Witten type conjecture. One has

$$\pmb{\mathrm{D}}^{0}_{m,2n}=\pmb{\Phi}_{k,m,2n},$$

and

$$\pmb{\mathrm{D}}^{N_f}_{m,2n}=2^{\frac{(N_f+2)k}{2}}\u000pmb{\Phi}^{N_f,c,1}_{k,m,2n}.$$

This is the explicit coefficient-level relation expected from the physical low-energy effective theory; the source proves only a weak version for $N_f=2,3$, leaving the full identities conjectural.