Kubertin’s generalized Erdős–Ginzburg–Ziv conjecture in higher rank

Prove that for every prime power q and every rank r, the generalized Erdős–Ginzburg–Ziv constant satisfies s_{kq}(C_q^r)=(k+r)q−r for all integers k≥r, including the ranks and parameter ranges not resolved by the rank-three results.

Background

The standard lower bound gives s_{kq}(C_q^r)≥(k+r)q−r for the elementary homocyclic group C_q^r when q is a prime power. Kubertin’s conjecture asserts that this lower bound is always sharp once the multiplier k is at least the rank r.

The paper proves all remaining cases for rank r=3, including the difficult cases (p,k)=(2,3) and (3,4), but explicitly notes that the conjecture remains unresolved in higher rank. Thus the open portion concerns ranks at least four and any other parameter ranges not covered by existing results.