Uniform boundedness of covering complexity for a fixed connected reductive group

Determine whether the covering complexity \(C(V)\) is bounded over all irreducible rational representations \(V\) of every fixed connected reductive group \(G\), and, if so, prove the bound \(C(V)\leq \max(1,\dim G/B)\).

Background

The paper establishes lower bounds showing that, for a fixed connected reductive group, complexities of suitable families of irreducible representations are at least the dimension of the flag variety in the limit. It also proves uniform upper bounds for particular families, including the conjugation representations of $SL_n(\mathbb C)$ and the representations $\Sym^k\mathbb C^3$ of $SL_3(\mathbb C)$.

These results leave unresolved whether every fixed connected reductive group has a uniform complexity bound across all its irreducible rational representations, and whether the dimension of the flag variety gives the correct universal bound.