Injectivity of the integer point transform without dilation

Let $d4ab,d4ac\binom{}{}$ be rational polytopes in $d4ad^d$, and let $d709^*$ be the vector specified in the paper's algebraic-independence condition.

Integer point transform injectivity conjecture. If

$$\sigma_{\mathcal{P}}(\xi^*)=\sigma_{\mathcal{Q}}(\xi^*)$$

then $d4ab=d4ac$.

This would strengthen the stated main theorem by eliminating the dilation factor $k$, making evaluation of the integer point transform at the single vector $d709^*$ a complete invariant for rational polytopes. The source presents this as an open question.