Marian–Oprea–Pandharipande conjecture relating higher-rank Segre series and Verlinde series

Let $S$ be a nonsingular projective surface and let $V\to S$ be a vector bundle of rank $s=r+1$, with $r\ne0$. Let $V^{[n]}$ be the induced bundle on $S^{[n]}$, and let $A_1(z),\ldots,A_5(z)$ be the universal power series appearing in the higher-rank Segre generating formula. Let $a_r(w)$ and $b_r(w)$ be the two unknown Verlinde series. After the change of variables

$$z=-\frac{1}{r}t(1+t)^{-r},\qquad w=\frac{t(-r+(-r+1)t)^{r^2-1}}{(-r(1+t))^{r^2}},$$

Marian–Oprea–Pandharipande conjecture. The universal series are

$$A_1(z)=(-r)^{-r-1}(1+t)^{-r}(-r+(-r+1)t)^{r+1},$$ $$A_2(z)=(-r)^r(1+t)^{r-1}(-r+(-r+1)t)^{-r},$$ $$A_3(z)=(-r)^{r^2}(1+t-rt)^{-1}(1+t)^{(r-1)^2}(-r+t(-r+1))^{-r^2},$$ $$A_4(z)=a_r(w),\qquad A_5(z)=b_r(w).$$

This refines the higher-rank extension of Lehn's formula and connects its last two universal series to the unknown Verlinde series. The supplied text does not establish the conjecture itself; the rank-one case is singular and is excluded by the stated formulas.