The Abel-map linearization conjecture for reflectionless KdV potentials

Let $\mathsf{E}$ be the spectral set, let $\mathcal{R}(\mathsf{E})$ denote the reflectionless potentials with spectrum $\mathsf{E}$, and let

$$\mathcal{B}\colon\mathcal{R}(\mathsf{E})\to\mathcal{D}(\mathsf{E})$$

be the divisor map. Let $\mathcal{A}\colon\mathcal{D}(\mathsf{E})\to\pi_1(\Omega)^*$ be the Abel map, and write $\alpha=\mathcal{A}(\mathcal{B}(V))=\{\alpha_k\}_{k\geq1}$. The Abel-map linearization conjecture. The composition map $\mathcal{A}\circ\mathcal{B}$ linearizes shifts of a reflectionless potential in space and time: there exists $\eta=\{\eta_k\}_{k\geq1}$ such that

$$\mathcal{A}(\mathcal{B}(V(x+x_0)))=\{\alpha_k-\eta_kx_0\bmod 1\}_{k\geq1},$$

and there exists $\eta^{(1)}=\{\eta_k^{(1)}\}_{k\geq1}$ such that

$$\mathcal{A}(\mathcal{B}(u(x,t)))=\{\alpha_k-\eta_k^{(1)}t\bmod 1\}_{k\geq1},$$

where $u(x,t)$ is the KdV solution with $u(x,0)=V(x)$. This is part of the proposed inverse-spectral/KdV program; the supplied text does not establish its resolution.