Finite Euler sums and symmetric Euler sums correspondence

Let $\bbN$ denote the positive integers. For each weight $w\in\mathbb{N}$, let $\mathsf{FES}_w$ be the $\mathbb{Q}$-vector space generated by finite Euler sums of weight $w$, and let $\mathsf{ES}_w$ be the $\mathbb{Q}$-vector space generated by Euler sums of weight $w$. For a signed index $\boldsymbol{\sl{s}}$, write $\zeta_\mathcal{A}(\boldsymbol{\sl{s}})$ for the corresponding finite Euler sum and $\zeta_\shuffle^{\mathcal S}(\boldsymbol{\sl{s}})$ for its shuffle-symmetric Euler sum. Finite Euler sums correspondence. There is an isomorphism

$$f_\mathsf{ES}: \mathsf{FES}_{w} \longrightarrow \displaystyle\!\frac{\mathsf{ES}_w}{\zeta(2)\mathsf{ES}_{w-2}}, \qquad \zeta_\mathcal{A}(\boldsymbol{\sl{s}})\longmapsto \zeta_\shuffle^{\mathcal S}(\boldsymbol{\sl{s}}).$$

This proposed isomorphism identifies finite Euler sums with shuffle-symmetric Euler sums modulo the subspace generated by multiplication by $\zeta(2)$. The source gives no resolution evidence for this assertion.