Generalisation of the Lagrangian subvariety proposition to (H,A)-stability

Let $Y$ be a bielliptic surface, let $f\colon X\to Y$ be its canonical covering of degree $n=2$, and let $w$ be a sheaf class on $Y$. Write $K_{X;f^*H,f^*A}(f^*w)$ and $K_{Y;H,A}(w)$ for the corresponding fibers of the second-Chern-class morphisms for $(H,A)$-stable sheaves. Generalisation to $(H,A)$-stability. If $f^*w$ is primitive, $f^*A$ is $f^*w$-general and $-\chi(f^*w,f^*w)\ge 6$, then $K_{X;f^*H,f^*A}(f^*w)$ is an irreducible symplectic manifold and the image of the morphism

$$K_{Y;H,A}(w)\stackrel{f^*}\to K_{X;f^*H,f^*A}(f^*w)$$

is a Lagrangian subvariety. This extends the preceding result from $H$-stability to $(H,A)$-stability; the source does not indicate whether the assertion has been proved or remains open.