The classification of exceptional special tuples for the weak Deligne-Simpson problem

Let $\kappa=2$, and consider tuples of conjugacy classes for which conditions i) and ii) of the source's genericity theorem hold, but for which the weak Deligne-Simpson problem is not solvable. A tuple is special if it is $l$-special for some $l$, as defined in the surrounding text. Exceptional-special-tuples conjecture. Special tuples are the only such tuples; equivalently, every tuple satisfying conditions i) and ii) with nonsolvable weak Deligne-Simpson problem is special. The preceding theorem establishes nonsolvability for special-diagonal tuples, and the source says that replacing “special-diagonal” by “special” appears plausible but is unproved; the proposed classification remains open.