Nathanson's conjecture on tiling intervals with two repeated gap lengths

Let $k$ and $l$ be positive integers, and let $p_1,\ldots,p_k,q_1,\ldots,q_l$ be positive integer gap lengths. A set of $k+l+1$ integers has gap sequence $p_1,\ldots,p_k,q_1,\ldots,q_l$ when its consecutive gaps are these values. Nathanson's conjecture. There is an interval of $\mathbb{Z}$ that can be partitioned into $(k+l+1)$-sets with the same gap sequence $p_1,\ldots,p_k,q_1,\ldots,q_l$ where $p_1=\cdots=p_k$ and $q_1=\cdots=q_l$. This conjecture extends the paper's focus from $4$-sets to arbitrary set sizes and asks for a partition using any prescribed gap sequence with two constant blocks of gaps; the source identifies it as an open question, and no resolution is given here.