Duality conjecture for recovery balanced codes

Let $\mathcal{C}$ be a linear code over a finite field, and let $\mathcal{C}^\perp$ denote its dual code. A code is recovery balanced when its random access expectation is equal to its dimension $k$.

Duality conjecture. A code $\mathcal{C}$ is recovery balanced if and only if its dual code $\mathcal{C}^\perp$ is recovery balanced.

The conjecture would extend the result proved for codes whose permutation automorphism group is transitive. It is supported by the fact that MDS, Hamming, and simplex codes are recovery balanced over every finite field, but the general closure of recovery balancedness under duality remains open.