Picard-number conjecture for moduli spaces of sheaves on a quadric surface

Let $X=D53F^1\timesD53F^1$ and let $M_{H_m}({\bf v})$ be the moduli space associated with a character ${\bf v}=(r,\nu,\Delta)\in K(X)$, where $r\geq 2$ and $\Delta\geq\frac{1}{2}$. Let $\epsilon\in\mathbb{Q}$ be sufficiently small depending on $r$, with $0<|\epsilon|\ll1$, and set $m=1+\epsilon$. Write $\operatorname{DLP}_{H_m}^{<r}(\nu)$ for the relevant DLP-surface, let $\lambda$ be the determinant-line-bundle map, and call ${\bf v}$ good or bad as in the paper. For an exceptional bundle $E$, write $[\overline E]$ for its associated class. Picard-number conjecture. The following hold: (1) If $\Delta>\operatorname{DLP}_{H_m}^{<r}(\nu)$, then

$$\operatorname{Pic}(M_{H_m}({\bf v}))\cong\mathbb{Z}^3$$

and $\lambda$ is an isomorphism. (2.a) If ${\bf v}$ is good, $\Delta=\operatorname{DLP}_{H_m}^{<r}(\nu)$, and a single exceptional bundle $E$ is associated to ${\bf v}$, then

$$\operatorname{Pic}(M_{H_m}({\bf v}))\cong\mathbb{Z}^2$$

and $\lambda$ is an epimorphism with $\ker\lambda\cong\mathbb{Z}[\overline E]$. (2.b) If ${\bf v}$ is bad under the same boundary and single-exceptional-bundle hypotheses, then

$$\operatorname{Pic}(M_{H_m}({\bf v}))\cong\mathbb{Z}$$

and $\lambda$ is an epimorphism with $\ker\lambda\supsetneq\mathbb{Z}[\overline E]$. (3.a) If $\Delta=\operatorname{DLP}_{H_m}^{<r}(\nu)>\frac12$ and at least two different exceptional bundles $E_1,E_2$ are associated to ${\bf v}$, then

$$\operatorname{Pic}(M_{H_m}({\bf v}))\cong\mathbb{Z}$$

and $\lambda$ is an epimorphism with $\ker\lambda\cong\mathbb{Z}[\overline{E_1}]+\mathbb{Z}[\overline{E_2}]$. (3.b) If $\Delta=\frac12$, then $M_{H_m}({\bf v})$ is a projective space and

$$\operatorname{Pic}(M_{H_m}({\bf v}))\cong\mathbb{Z}.$$

This conjecture predicts that the Picard number is determined by the position of ${\bf v}$ relative to the $\operatorname{DLP}_{H_m}^{<r}$-surface and by whether ${\bf v}$ is good or bad. The paper notes that the conjecture is open; in particular, the second generator of $\ker\lambda$ in the bad boundary case is not explicitly described.