On the dimension of spaces of solutions of the noncommutative unitary sigma-model

There is a long-standing conjecture that the complex dimension of the space of all solutions (of the model in the title) having normalized energy $e$, canonical rank $r$ and minimal uniton number $u$, is equal to $r$. We plan to give a proof of (a sharpened version of) this conjecture in the case when $u=2$ and discuss related topics.