Isomorphism conjecture for the zeta and lambda realizations

For $n\in\mathbb{N}^*$ and $\Sigma\in\mathfrak{S}$, let $\boldsymbol{C}_{n,\Sigma}$ be the corresponding space of composition arrays, and let $\mathcal{Z}_{n,\Sigma}$ and $\mathcal{L}_{n,\Sigma}$ be the associated spaces of multiple zeta values and lambda values. The maps $\zeta_A$ and $\lambda_A$ are $\mathbb{F}_p$-linear maps from $\boldsymbol{C}_{n,\Sigma}$ to these spaces. Isomorphism conjecture. The maps define isomorphisms of vector spaces

$$\mathcal{Z}_{n,\Sigma}\xleftarrow{\zeta_A}\boldsymbol{C}_{n,\Sigma}\xrightarrow{\lambda_A}\mathcal{L}_{n,\Sigma}.$$

This conjecture asserts that the composition-array model simultaneously gives unique descriptions of both families of values; the source gives no resolution.