McNamara–van Willigenburg's factorization conjecture for equivalent skew diagrams

Let $\mathcal{D}$ and $\mathcal{E}$ be skew diagrams, and let $\mathcal{D}^*$ denote the diagram obtained from $\mathcal{D}$ by rotation through 180 degrees. Write $\mathcal{E}_i=W_iO_iW_i$ when this decomposition is applicable, and let $\circ_{W_i}$ denote the operation used to combine skew diagrams along $W_i$. Two skew diagrams are equivalent when $\mathcal{D}\sim\mathcal{E}$, meaning that their skew Schur functions satisfy $s_{\mathcal{D}}=s_{\mathcal{E}}$. For some $r$, suppose

$$\mathcal{D}=(\cdots((\mathcal{E}_1\circ_{W_2}\mathcal{E}_2)\circ_{W_3}\mathcal{E}_3)\cdots)\circ_{W_r}\mathcal{E}_r$$

and

$$\mathcal{E}=(\cdots((\mathcal{E}'_1\circ_{W'_2}\mathcal{E}'_2)\circ_{W'_3}\mathcal{E}'_3)\cdots)\circ_{W'_r}\mathcal{E}'_r.$$

McNamara–van Willigenburg's conjecture. The diagrams $\mathcal{D}$ and $\mathcal{E}$ satisfy $\mathcal{D}\sim\mathcal{E}$ if and only if the factors satisfy: $\mathcal{E}_1,\ldots,\mathcal{E}_r$ are skew diagrams; for $2\leq i\leq r$, $\mathcal{E}_i=W_iO_iW_i$ satisfies Hypotheses I--IV; $\mathcal{E}'_1=\mathcal{E}_1$ or $\mathcal{E}'_1=\mathcal{E}_1^*$; and for $2\leq i\leq r$, $(\mathcal{E}'_i,W'_i)=(\mathcal{E}_i,W_i)$ or $(\mathcal{E}'_i,W'_i)=(\mathcal{E}_i^*,W_i^*)$. Moreover, the equivalence class of $\mathcal{D}$ contains $2^\kappa$ elements, where $\kappa$ counts factors $\mathcal{E}_i$ in any irreducible factorization of $\mathcal{D}$ for which $\mathcal{E}_i\neq\mathcal{E}_i^*$. This conjecture would characterize equivalence classes of skew diagrams using the sufficient conditions previously established by McNamara and van Willigenburg; the source gives no evidence of resolution, so its status remains open.