The multiple-harmonic-sum analogue of the Kontsevich–Zagier period conjecture

Fix $N>1$ and work on $\mathbb{P}^{1}-\{0,\mu_N,\infty\}$. Let $p$ range over primes not dividing $N$, let $\alpha\in\mathbb{N}$, and let $\operatorname{har}_{p^{\alpha}}$ denote the corresponding prime multiple harmonic sums. Multiple-harmonic-sum period conjecture. Any equality between prime multiple harmonic sums $\operatorname{har}_{p^{\alpha}}$ that is valid either for all primes $p$ prime to $N$, or for infinitely many $\alpha\in\mathbb{N}$ in $\prod_p\mathbb{Q}_p$, can be transformed from one side to the other using the stated operations on the stated algebra of functions, simultaneously for all such $p$, respectively for all such $\alpha$. This is proposed as a Kontsevich–Zagier-type completeness principle for iterated series; the source gives no resolution.