Branching component graph product conjecture for horizontally concatenated partitions

Let $\lambda^{(1)}$ and $\lambda^{(2)}$ be partitions. Write $\ell(\lambda^{(2)})$ for the number of parts of $\lambda^{(2)}$ and $m_{1}^{(1)}$ for the multiplicity of $\lambda^{(1)}_{1}$ in $\lambda^{(1)}$. Suppose that $\ell(\lambda^{(2)})\leq m_{1}^{(1)}-2$. Let $\lambda^{(1)}\lambda^{(2)}$ be their horizontal concatenation, defined by

$$(\lambda^{(1)}\lambda^{(2)})_{i}=\lambda^{(1)}_{i}+\lambda^{(2)}_{i}.$$

Branching component graph product conjecture. The branching component graph satisfies

$$\mathcal{BC}(\lambda^{(1)}\lambda^{(2)})=\mathcal{BC}(\lambda^{(1)})\times\mathcal{BC}(\lambda^{(2)}),$$

where $\times$ is the directed graph product, and the vertex $(\mu^{(1)},\mu^{(2)})$ is labeled by $\mu^{(1)}\mu^{(2)}$. The surrounding discussion states that the vertex and weight assertions are proved, while the edge properties are conjectural; this product statement is included as part of that conjectural structure.