Pinheiro–Machado–Firer extension conjecture for partition-induced combinatorial metrics

Let $\mathbb{F}_2$ be the binary field and let $\mathbf{H}=\mathbb{F}_2^\Omega$. For $k\in[1,|\Omega|]$, write $\mathcal{P}(k,\Omega)$ for the covering of $\Omega$ by all $k$-element subsets, and let $\mathcal{CO}(\mathbf{H},\mathcal{P}(k,\Omega))$ be the partition by $\mathcal{P}(k,\Omega)$-weight. Pinheiro–Machado–Firer conjecture. For every additive code $C\leqslant\mathbf{H}$ and every $f\in\operatorname{Hom}(C,\mathbf{H})$ satisfying

$$\operatorname{wt}_{\mathcal{P}(k,\Omega)}(\alpha)=\operatorname{wt}_{\mathcal{P}(k,\Omega)}(f(\alpha))\quad\text{for all }\alpha\in C,$$

there exists $\varphi\in\operatorname{Aut}(\mathbf{H})$ such that $\varphi|_C=f$ and

$$\operatorname{wt}_{\mathcal{P}(k,\Omega)}(\alpha)=\operatorname{wt}_{\mathcal{P}(k,\Omega)}(\varphi(\alpha))\quad\text{for all }\alpha\in\mathbf{H}.$$

Equivalently, for every $k\in[1,|\Omega|]$, the partition $\mathcal{CO}(\mathbf{H},\mathcal{P}(k,\Omega))$ satisfies the MacWilliams extension property (MEP). This conjecture concerns whether every weight-preserving homomorphism on an additive code extends to a global weight-preserving automorphism; its resolution is not supplied in the source.