The exponential valuation conjecture for real polytopes

Let $n$ be a positive integer, let $\mathcal{P}^n$ denote the family of polytopes in $\mathbb{R}^n$, and let $\mathcal{M}(\mathbb{R}^n)$ denote the space of real-valued functions on $\mathbb{R}^n$. A valuation $Z:\mathcal{P}^n\to\mathcal{M}(\mathbb{R}^n)$ is translatively exponential and $\operatorname{GL}(n,\mathbb{R})$ covariant if it satisfies the corresponding translation and linear-covariance identities. Exponential valuation conjecture. Any translatively exponential and $\operatorname{GL}(n,\mathbb{R})$ covariant valuation

$$Z:\mathcal{P}^n\to\mathcal{M}(\mathbb{R}^n)$$

is of the form $Z=c\mathcal{L}$ for a constant $c\in\mathbb{R}$, where $\mathcal{L}$ is the Laplace transform. The analogous classification is known for continuous valuations on compact convex sets, whereas the claim for valuations on arbitrary polytopes is presented as a belief and remains open.