Manageable description of the cokernel controlling the first shifted twisted Witt group of a real surface

Characterize the cokernel of the connecting homomorphism \partial_L:H^0(X,\overline{I})\to H^1(X,I^2(L)) for a smooth real surface X and line bundle L, in a manageable general form that accounts for its interaction with H^1(X,K^2(L)).

Background

For a smooth real surface X and line bundle L, the paper studies W^1(X,L) through the exact sequence involving H^1(X,I^2(L)), the connecting homomorphism \partial_L, and the kernel of the twisted Steenrod square \Sq_L^2. A purely topological description analogous to the one available for the top shifted Witt group fails because the image of \partial_L can intersect H^1(X,K^2(L)) nontrivially.

The paper gives an explicit product-of-elliptic-curves example in which this intersection is nontrivial, showing that the obstruction cannot generally be captured solely by the real-locus topology. It nevertheless leaves unresolved whether the resulting cokernel admits a manageable description in general.