Uniform bounded complexity for irreducible representations of SL3(C)

Determine whether there exist constants \(\theta<1\) and \(C_0\) such that, for every pair of nonnegative integers \((a,b)\) and every Borel-stable subspace \(W_0\leq V(a,b)\) of the irreducible \(SL_3(\mathbb C)\)-representation \(V(a,b)\) satisfying \(\dim W_0\geq \theta\dim V(a,b)\), one has \(\diamp[SL_3(\mathbb C)](V(a,b),W_0)\leq C_0\).

Background

The paper proves that the covering complexity of the representations $\Sym^k\mathbb C^3$ of SL3(C)SL_3(\mathbb C) is bounded uniformly in kk. The proof reduces the problem to bounding the diameter of sufficiently large Borel-stable subspaces and then exploits the geometry of plane curves and multiplicity conditions on grids.

The authors state that their argument does not extend to the other irreducible SL3(C)SL_3(\mathbb C)-representations V(a,b)V(a,b). A positive answer to this question would provide the missing bounded-diameter input and extend the bounded-complexity result from the symmetric-power family to all irreducible representations of SL3(C)SL_3(\mathbb C).

References

We have not been able to extend the argument to the other irreducible representations V(a,b) of SL_3(C). The same method would work provided one could show that large Borel-stable subspaces of V(a,b) have bounded diameter. Are there \theta < 1 and C_0 such that for every pair (a,b) and every Borel-stable subspace W_0 \leq V(a,b) with \dim W_0 \geq \theta \dim V(a,b) one has \diampSL_3(C) \leq C_0?

— Additive diameters and covering complexity of irreducible representations  (2609.03882 - Jezernik et al., 3 Sep 2026) in Question 2, Section 8, subsection “Questions”