MacWilliams identity conjecture for convolutional-code adjacency matrices

Let ${\mathcal C}$ be a convolutional code over ${\mathbb F}$ with adjacency matrix $\Lambda$, and let $\widehat{\mathcal C}$ be its dual code with generalized adjacency matrix $\widehat\Lambda=(\widehat\lambda_{X,Y})$. Let $q=|{\mathbb F}|$, let $k$ be the dimension of ${\mathcal C}$, let $\delta$ be their common degree, let ${\mathcal H}$ be the MacWilliams matrix, and let \text{{\text{\textbf H}}} denote the corresponding entrywise transformation. For connected state pairs $(X,Y)\in{\mathcal F}$, there exists $P\in \operatorname{GL}_{\delta}({\mathbb F})$.

MacWilliams identity conjecture. The matrix

$$q^{-k}\text{${\text{\textbf H}}$}\big({\mathcal H}\Lambda^t{\mathcal H}^{-1}\big)$$

where \text{{\text{\textbf H}}} is applied entrywise, is a representative of the generalized adjacency matrix of $\widehat{\mathcal C}$; equivalently,

$$\widehat\lambda_{X,Y}=q^{-k}\text{${\text{\textbf H}}$}\big(({{\mathcal H}\Lambda^t{\mathcal H}^{-1}})_{XP,YP}\big) \quad\text{for all }(X,Y)\in{\mathcal F}.$$

This conjecture would extend the classical MacWilliams identity from block codes to convolutional codes by showing that the adjacency matrix of a code determines that of its dual through an explicit transformation. The surrounding section proves the identity for a particular class of codes, while the displayed assertion is formulated as the general conjectural statement.