Universal-series correspondence for Hilbert schemes and strange duality

Let $S$ be a surface and let $V_s(w),W_s(w),X_s(w),Y_s(w),Z_s(w)$ be the universal power series associated with a class in $K_0(S)$ of rank $s$. Let $g_r(z),f_r(z),A_r(z),B_r(z)$ be the universal series from the corresponding Hilbert-scheme formulas. Define

$$\phi_s(w)=V_s(w)^{2-s},\qquad w=z\phi_s(w),\qquad s=r+1.$$

Universal-series correspondence. The series satisfy

$$g_r(z)=V_s(w)\cdot W_s(w),$$ $$f_r(z)=\frac{X_s(w)}{\phi_s(w)^4\left(\frac{dz}{dw}\right)^2},$$ $$A_r(z)=Y_s(w),\qquad B_r(z)=Z_s(w).$$

These identities express the relationship between the universal series governing Euler characteristics and those governing top Chern classes of tautological bundles, as suggested by Le Potier's strange duality. The source does not provide evidence that the identities have been proved, so their resolution remains open.