The period conjecture for reduction of $p$-adic multiple zeta values

Let $\operatorname{red}_{\zeta}$ be the map obtained by reducing $p$-adic multiple zeta values modulo sufficiently large primes, from the $\mathbb{Q}$-algebra of $p$-adic multiple zeta values in $\prod_p\mathbb{Z}_p$ onto the $\mathbb{Q}$-algebra of finite multiple zeta values. Reduction period conjecture. The map

$$\operatorname{red}_{\zeta}$$

is an isomorphism. The source presents this as the remaining period-theoretic component of Kaneko–Zagier's conjecture and gives no resolution.