String-space description conjecture for Bott–Samelson Okounkov bodies

Fix a Borel subgroup $B\subset G$ and a sequence $I=(i_1,\ldots,i_\ell)$ of simple roots defining the Bott–Samelson variety $R_I$. Let $P_v(\mathcal L)$ be the Okounkov body of a line bundle $\mathcal L$ on $R_I$ with respect to the valuation defined by the flag of subsequence embeddings. Let $\mathcal S_I\subseteq\mathbb R^\ell$ be the associated string space, let $D_i$ be the corresponding divided difference operators, let $E_u$ denote translation by $u\in\mathbb R^\ell$, and let $p:\mathbb R^\ell\to\mathbb R^r$ be the projection associated with the string-space decomposition. String-space description conjecture. For every line bundle $\mathcal L$ on $R_I$, there exists a point $\mu\in\mathbb R^r$ and vectors $u_1,\ldots,u_\ell\in\mathbb R^\ell$ such that

$$P_v(\mathcal L)=E_{u_1}D_{i_1}E_{u_2}D_{i_2}\ldots E_{u_\ell}D_{i_\ell}(a_\mu)$$

for any point $a_\mu\in\mathbb R^\ell$ satisfying $p(a_\mu)=\mu$. This would give an explicit description of Okounkov bodies on Bott–Samelson varieties in terms of the associated string-space operators; the supplied text does not indicate whether the conjecture has been proved or disproved.