Periodic optimality for one-dimensional compact forbidden-distance sets

Let $K\subset[1,\infty)$ be a compact set with $1\in K$. A sphere packing is $K$-admissible if all distances between distinct centers avoid $K$ in the paper's normalization, and a periodic packing is one whose center set is a finite union of translates of a lattice. Periodic-optimality conjecture. There exists a $K$-admissible periodic sphere packing of $\mathbb{R}$ with maximal density. The paper proves this under additional hypotheses on the accumulation points of $K$, but leaves the assertion for arbitrary compact $K$ open.