Towards a dichotomy for the switch list homomorphism problem for signed graphs

We make advances towards a structural characterisation of the signed graphs $H$ for which the list switch $H$-colouring problem $\mathrm{LSwHom}(H)$ problem is polynomial time solvable. We conjecture a characterisation for signed graphs that can be switched to graphs such that every negative edge is also positive, and prove the characterisation in the case that the signed graph is reflexive.