Injectivity of period maps for convergent motivic multiple harmonic values

Let $I\subset\mathcal{P}\times\mathbb{N}$, and let $\widehat{\operatorname{per}}_{\operatorname{har}_I}$ be the period map on weight-adically convergent sums of motivic multiple harmonic values. Injectivity conjecture for motivic multiple harmonic values. If $I$ contains a set of the form $p^{\mathbb{N}^{\ast}}$ for a prime $p$, or a set of the form $\mathcal{P}^{\alpha}$ for some $\alpha\in\mathbb{N}^{\ast}$, then $\widehat{\operatorname{per}}_{\operatorname{har}_I}$ is injective. Equivalently, every relation expressing the vanishing of a weight-adically and $p$-adically convergent sum of multiple harmonic values lifts to infinitely many equalities among motivic multiple zeta values. This is proposed as a lifting principle for relations among harmonic values.