Commutativity of definable sums over discrete sets

Let $\mathbb{K}$ be an expansion of an ordered field, let $D \subseteq K_{\geq 0}$ be a definable closed discrete subset, let $h:D\to K_{\geq 0}$ be definable, and let $\sigma:D\to D$ be a definable bijection. Then the commutativity conjecture for definable sums.

$$\sum_{d\in D}h(d)=\sum_{d\in D}h(\sigma(d)).$$

That is, if the sum on the left exists, then the sum on the right also exists and has the same value. The paper notes that the corresponding result is proved under the stronger assumption that $\mathbb{K}$ is unrestrained, while it cannot prove the result in general without assuming that the structure defines a discrete subring.