The exact-sequence conjecture for totally odd motivic multiple zeta values

For each $r\geq 3$, let $gr_r^{\mathfrak{D}}\mathcal{A}^{odd}$ denote the depth-graded totally odd part modulo $\zeta^{\mathfrak{m}}(2)$, let $\mathbb{P}^{\vee}$ be the weight-dual of the restricted even period-polynomial space, and let

$$\widetilde{\partial}:gr_r^{\mathfrak{D}}\mathcal{A}^{odd}\longrightarrow gr_1^{\mathfrak{D}}\mathcal{A}^{odd}\otimes gr_{r-1}^{\mathfrak{D}}\mathcal{A}^{odd},$$ $$D:gr_1^{\mathfrak{D}}\mathcal{A}^{odd}\otimes gr_{r-1}^{\mathfrak{D}}\mathcal{A}^{odd}\longrightarrow \mathbb{P}^{\vee}\otimes gr_{r-2}^{\mathfrak{D}}\mathcal{A}^{odd}$$

be the maps constructed in the paper. Exact-sequence conjecture. For $r\geq3$, there is an exact sequence

$$0\longrightarrow gr_r^{\mathfrak{D}}\mathcal{A}^{odd}\xrightarrow{\widetilde{\partial}}gr_1^{\mathfrak{D}}\mathcal{A}^{odd}\otimes gr_{r-1}^{\mathfrak{D}}\mathcal{A}^{odd}\xrightarrow{D}\mathbb{P}^{\vee}\otimes gr_{r-2}^{\mathfrak{D}}\mathcal{A}^{odd}\longrightarrow0.$$

The sequence generalizes the established depth-two exact sequence and is intended to explain the structure predicted by the uneven-part conjecture. The paper does not report a resolution.