The Furusho conjecture on algebraic relations between real and p-adic multiple zeta values

Let $p$) be a prime, and let $\zeta(\mathfrak{s})$ and $\zeta_p(\mathfrak{s})$ denote the real-valued and $p$-adic multiple zeta values associated with an admissible index $\mathfrak{s}$. For $\mathfrak{Z}$ and $\mathfrak{Z}_p$, write the $\mathbb{Q}$-vector spaces spanned by $1$ and all real-valued and $p$-adic multiple zeta values, respectively. Furusho's conjecture. For every prime $p$, if

$$f(\zeta(\mathfrak{s}_1),\ldots,\zeta(\mathfrak{s}_m))=0$$

for $f\in\mathbb{Q}[X_1,\ldots,X_m]$, then

$$f(\zeta_p(\mathfrak{s}_1),\ldots,\zeta_p(\mathfrak{s}_m))=0.$$

Equivalently, the assignment $\zeta(\mathfrak{s})\mapsto\zeta_p(\mathfrak{s})$ should define a surjective $\mathbb{Q}$-algebra homomorphism $\mathfrak{Z}\twoheadrightarrow\mathfrak{Z}_p$. This conjecture seeks an explicit connection between the algebraic relations of real-valued and $p$-adic multiple zeta values; the paper notes evidence from regularized double-shuffle relations and results of Furusho and Jafari, but does not state a resolution.