Generic strange duality for moduli spaces on elliptic surfaces

Let $X$ be a simply connected elliptic surface with a section and at worst nodal fibers, and let $v$ and $w$ be orthogonal topological types of ranks $r,s\geq 2$ satisfying

$$c_1(v)\cdot f=c_1(w)\cdot f=1,$$

and

$$\dim {\mathfrak M}_v+\dim {\mathfrak M}_w\geq \Delta,$$

where

$$\Delta=\chi(X,\mathcal O_X)\left((r+s)^2+(r+s)+2\right)-2(r+s).$$

Assume the polarization is suitable, and let $\Theta_v$ and $\Theta_w$ be the theta line bundles on the corresponding moduli spaces. Generic strange duality conjecture. Under these assumptions,

$$\mathsf D:H^0({\mathfrak M}_v,\Theta_w)^{\vee}\to H^0({\mathfrak M}_w,\Theta_v)$$

is an isomorphism. The conjecture is established up to showing that the birational identifications of ${\mathfrak M}_v$ and ${\mathfrak M}_w$ with Hilbert schemes of points hold away from codimension $2$.