Maximum size of symmetric orthogonal Sudoku-hypercube families

Determine whether the maximum size of a family of mutually orthogonal Sudoku hypercubes of order q^3 with block size (q^2,q^2,q^2) equals q^2(q^2−1)^2 for every prime power q, and prove the asserted equality if so.

Background

For block sizes (q^3,q^2,q) and (q^3,q^3,1), Theorem 4.3 establishes exact maximum family sizes using the general upper bound from Lemma 4.1. For the symmetric block size (q^2,q^2,q^2), the paper constructs q^2(q^2−1)(q^2−q) mutually orthogonal Sudoku hypercubes.

Lemma 4.1 gives the upper bound q^2(q^2−1)^2, so the exact maximum is unresolved. The constructed family is asymptotically optimal, falling short of the conjectured value by the factor (q+1)/q.