  Let $d$ be a positive integer. For a finite set $X \subseteq \mathbb{R}^d$,
we define its integer cone as the set $\mathsf{IntCone}(X) := \{ \sum_{x \in X}
\lambda_x \cdot x \mid \lambda_x \in \mathbb{Z}_{\geq 0} \} \subseteq
\mathbb{R}^d$. Goemans and Rothvoss showed that, given two polytopes
$\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^d$ with $\mathcal{P}$ being
bounded, one can decide whether $\mathsf{IntCone}(\mathcal{P} \cap
\mathbb{Z}^d)$ intersects $\mathcal{Q}$ in time
$\mathsf{enc}(\mathcal{P})^{2^{\mathcal{O}(d)}} \cdot
\mathsf{enc}(\mathcal{Q})^{\mathcal{O}(1)}$ [J. ACM 2020], where
$\mathsf{enc}(\cdot)$ denotes the number of bits required to encode a polytope
through a system of linear inequalities. This result is the cornerstone of
their XP algorithm for BIN PACKING parameterized by the number of different
item sizes.
  We complement their result by providing a conditional lower bound. In
particular, we prove that, unless the ETH fails, there is no algorithm which,
given a bounded polytope $\mathcal{P} \subseteq \mathbb{R}^d$ and a point $q
\in \mathbb{Z}^d$, decides whether $q \in \mathsf{IntCone}(\mathcal{P} \cap
\mathbb{Z}^d)$ in time $\mathsf{enc}(\mathcal{P}, q)^{2^{o(d)}}$. Note that
this does not rule out the existence of a fixed-parameter tractable algorithm
for the problem, but shows that dependence of the running time on the parameter
$d$ must be at least doubly-exponential.
