The strengthened 2-criticality conjecture for row-isotoped abelian 2-groups

Let $L_s$ be the latin square of order $n=2^s$, and let $\alpha_{k,k'}$ be the row isotopism that interchanges rows $k$ and $k'$ and fixes every other row. Suppose that

$$\left|k-k'\right|<3.$$

Strengthened 2-criticality conjecture. The set $\operatorname{gcs}(\alpha_{k,k'}L_s)$ is 2-critical and strong, and completes top down to $\alpha_{k,k'}L_s$.

The theorem preceding this conjecture proves the claim under the additional hypothesis that $k$ and $k'$ lie among four consecutive indices beginning at a multiple of $4$. The authors expect that this stronger version remains true when that additional condition is weakened, but no resolution is given here.