Elliptic identity lifting conjecture for higher-genus multiple zeta values

Let $l,k_1,\ldots,k_l\geq1$ and let $n_{1,1},\ldots,n_{k_1,1},\ldots,n_{1,l},\ldots,n_{k_l,l}\geq1$. Let $\omega^{\mathrm{ell}}$ denote elliptic multiple zeta values and let $I$ be an identity satisfying

$$I(\omega^{\mathrm{ell}}(n_{1,1}-1,\ldots,n_{k_1,1}-1),\ldots,\omega^{\mathrm{ell}}(n_{1,l}-1,\ldots,n_{k_l,l}-1))=0.$$

Elliptic identity lifting conjecture. For any $j\in\{1,\ldots,h\}$ and any genus $h\geq1$, the corresponding higher-genus multiple zeta values satisfy

$$I\!\left(\zeta_{\mathfrak A_j}\!\left(j^{n_{1,1}},\ldots,j^{n_{k_1,1}}\right),\ldots,\zeta_{\mathfrak A_j}\!\left(j^{n_{1,l}},\ldots,j^{n_{k_l,l}}\right)\right)=0,$$

with appropriate normalization of the appearing genus-zero multiple zeta values. The conjecture proposes that known identities among elliptic multiple zeta values lift to the corresponding higher-genus values despite the additional sigma terms. The paper reports numerical tests supporting the claim, but no general proof is known.