Indecomposable images of symmetric band modules under the unfolding functor

Determine whether the necessary similarity condition [?] [?] is sufficient, over an arbitrary ground field K, for a folded gentle algebra band module to occur as an indecomposable image under the unfolding functor U of a symmetric band module; specifically, determine whether [?] [?] holds whenever the band module M(w,m,[?]) is the image of such a symmetric band module.

Background

The paper studies folded gentle algebras through an unfolding functor U from the module category of a folded gentle algebra to the module category of a corresponding unfolded gentle algebra. Symmetric band modules are the most technically complicated objects in this comparison because their unfolded images can either remain indecomposable or split into two indecomposable band modules.

Proposition \ref{prop:USymBand} gives a necessary similarity condition for a band module M(w,m,\phi) over the unfolded gentle algebra to occur as an indecomposable image of a symmetric band module: \phi must be similar to a scalar multiple of its inverse, namely \mu_w^2\lambda_2\lambda'_2\phi^{-1}. The authors observe that this condition appears sufficient when K=\mathbb{R}, but do not establish sufficiency over general fields; resolving this would characterize precisely when symmetric band modules have indecomposable images under U.