Euler's 36 officers problem has been in the news recently, following the passing of the last of 'Euler's Spoilers'.

It notoriously has no solution for the 6x6 square with 6 officers (colonel, lieutenant-colonel, major, captain, lieutenant, and sub-lieutenant) and 6 regiments.

One possible modification, while still keeping 6 regiments, would be to allow substitution of a junior officer (or officers) in one or more of the regiments by a new rank - say, Sensitivity Counsellor, or SC - more reflective of the needs and aspirations of a modern-day military.

What is the minimum number of SC's needed to make the problem solvable?