Fine Grained Complexity for VAS Boundedness

A ($d$-dimensional) Vector Addition System is a set of vectors $T \subseteq \mathbb Z^d$ and induces a single-step transition relation $\mathrm\rightarrow\ \subseteq \mathbb N^d \times \mathbb N^d$ by $\mathbf u \rightarrow \mathbf v \Leftrightarrow \mathbf v = \mathbf u + \mathbf t$ for some $\mathbf t \in T$. The transitive closure of $\rightarrow$, $\rightarrow^*$, is called the reachability relation and for some vector $\mathbf u \in \mathbb N^d$, we call $R(\mathbf u) = \{\mathbf v \mid \mathbf u \rightarrow^* \mathbf v\}$ the reachability set of $\mathbf u$. The boundedness problem for VAS asks if, given a vector $\mathbf u$, the set $R(\mathbf u)$ is finite. It has been long known that the boundedness problem for VAS is EXPSPACE-complete. A more careful analysis reveals an upper bound of $\mathcal O(n^{2^{d \log d}})$. A similar upper bound for VAS coverability was recently tightened to $\mathcal O(n^{2^d})$. Can we tighten the upper bound for boundedness too?