Canonical-divisor formula for Hilbert-scheme virtual classes

Let $V$ be a surface with $p_g(V)>0$, let $K$ be an effective canonical divisor, and fix $m\in H^2(V,\mathbb{Z})$. Let

$$C(m,K)=\sum_{(D_1,D_2)\in\alpha^{-1}(K)}(-1)^{h^0(\mathcal{O}_{D_1}(D_1))}l(D_1,D_2)[D_1,D_2]$$

be the zero-cycle on $\operatorname{Hilb}^{m}_V\times_{\operatorname{Pic}^{m}_V}\operatorname{Hilb}^{k-m}_V$, and let $p_1,p_2$ be its projections to the two Hilbert schemes. Canonical-divisor virtual-class conjecture. One has

$$[[\operatorname{Hilb}^m_V]]=(p_1)_*C(m,K),$$

and

$$[[\operatorname{Hilb}^{k-m}_V]]=(-1)^{\chi(\mathcal{O}_V)+\frac{m(m-k)}{2}}(p_2)_*C(m,K).$$

This is presented as the strongest algebro-geometric analogue hoped for after Witten’s formula for the Seiberg–Witten invariant; the source gives no proof or resolution.