Conjectural Verlinde–Segre correspondence for Hilbert schemes

Let $S$ be a smooth projective surface, let $r$ be the rank parameter, and let $A_3(x),A_4(x)$ and $B_3(t),B_4(t)$ denote the universal power series in the Segre and Verlinde series, respectively. Fix $k=r+1$. Under the changes of variables

$$x=s(1-rs)^{-r},$$ $$t=\frac{s(1-(r-1)s)^{r^2-1}}{(1-rs)^{r^2}},$$

Verlinde–Segre correspondence. The universal series satisfy

$$A_3(x)=B_3(t),\qquad A_4(x)=B_4(t).$$

This correspondence is motivated by Le Potier's strange duality conjecture and relates Segre and Verlinde generating series for Hilbert schemes. The stated assertion is conjectural; the supplied source gives no resolution status.