Towards Harder-Narasimhan filtrations for Fukaya-Seidel type categories with coefficients

We report on progress in our joint work with F. Haiden, L. Katzarkov, and P. Pandit on a program of extending the Bridgeland-Smith construction of stability conditions to the case of $SL(3)$ spectral curves. We consider Fukaya-Seidel categories of graph Lagrangians with coefficients in a constant category, in our case of type $A2-CY2$, on a contractible flat Riemann surface. As in the recent theory of "perverse schobers", objects involve putting triangles at the threefold vertices of the underlying graph. The main question for defining a stability condition is to try to construct a deformation of an object to bring out the first step of its Harder-Narasimhan filtration. We sketch our current view of this process and discuss some of the remaining unsolved problems.