Covering-radius conjecture for lattice polytopes with prescribed interior points

Let $k,d\in\mathbb{N}$ be nonnegative integers with $k\geq2$. For every lattice $d$-polytope $P$ with $k$ interior lattice points, the prescribed-interior-points conjecture.

$$\mu(P)\leq\frac{d-1}{2}+\frac{1}{k+1},$$

with equality if and only if $P$ is obtained by direct sums and/or translations of a segment $[0,k+1]$ and one or more standard terminal simplices $S(\mathbf{1}_l)$. This extends the proposed extremal description for non-hollow lattice polytopes to a fixed number of interior lattice points. The source presents it as a natural conjecture without a resolution.