The alternating Euler-sum permutation conjecture for unramified motivic sums

Let $a\in\mathbb{N}_0$, $\ell\in\mathbb{N}$, and let $u_j,v_j\in X:=\{1,\bar2,3,\bar4,\dots\}$ for $j=1,\dots,\ell$, with

$$1\notin\{u_j:1\leq j\leq\ell\}\displaystyle\!\cap\{v_j:1\leq j\leq\ell\}.$$

For $\mathfrak S_\ell$ the symmetric group on $\ell$ letters, the motivic Euler sums $\zeta_a^\fm(\cdots)$ are understood in the notation of the paper. The unramified permutation-sum conjecture. Both sums

$$\displaystyle\!\sum_{\sigma,\tau\in\mathfrak S_\ell}\zeta_a^\fm(u_{\sigma(1)},v_{\tau(1)},\dots,u_{\sigma(\ell)},v_{\tau(\ell)})$$

and

$$\displaystyle\!\sum_{\sigma\in\mathfrak S_\ell,\,\tau\in\mathfrak S_{\ell-1}}\zeta_a^\fm(u_{\sigma(1)},v_{\tau(1)},\dots,v_{\tau(\ell-1)},u_{\sigma(\ell)})$$

are unramified. For $a=0$, applying the period map reduces this to a conjecture of M. Hirose and N. Sato; the stated motivic families extend the preceding proved unramified families, but the general assertion remains open.