The depth-one realization conjecture for the weight-five Grassmannian polylogarithm

Let $\operatorname{Gr}_5$ denote the weight-five Grassmannian polylogarithm, let $\operatorname{Alt}_{10}$ denote alternation over ten points, and let $\operatorname{r}_3^\pm$ be the invariant and anti-invariant combinations of the triple ratio under swapping its two triples. Consider the combination

$$\frac{1}{640}\,\mathrm{Gr}_5+\frac{5}{3}\operatorname{Alt}_{10}\left[2I_{4,1}(\operatorname{cr}(136|2459),\operatorname{r}_3^+(12|345,678))+I_{4,1}(\operatorname{cr}(346|1279),\operatorname{r}_3^-(12|345,678))\right].$$

Weight-five depth-one conjecture. (i) There exists a formal linear combination $\mathcal{Q}_5(v_1,\dots,v_{10})$ of rational functions on $\operatorname{Conf}_{10}(5)$ such that $\operatorname{Li}_5(\mathcal{Q}_5(v_1,\dots,v_{10}))$ equals the displayed combination modulo products. (ii) Assuming (i), the function $\varphi\colon\operatorname{GL}_5(\mathbb{C})^{10}\to\mathbb{R}$ defined by

$$\varphi(g_1,\dots,g_{10})=\mathcal{L}_5(\mathcal{Q}_5(g_1v,\dots,g_{10}v))$$

is a bounded measurable $9$-cocycle whose continuous cohomology class is a non-zero rational multiple of the Borel class $b_5^{(5)}\in H^9_{\mathrm{cts}}(\operatorname{GL}_5(\mathbb{C}),\mathbb{R})$. This is motivated by the conjectured structure of the motivic Lie coalgebra in weight five: the corrected Grassmannian expression is expected to admit a depth-one description and realize the Borel regulator class, but the existence of $\mathcal{Q}_5$ and the subsequent cohomological identification are not established here.