Sparsest-obstruction conjecture for integer pinwheel scheduling

Let $A=(a_i)_{i\in[k]}$ be an integer pinwheel scheduling instance, with periods arranged as $a_1\leq\cdots\leq a_k$, and let

$$\mathrm D(A)=\sum_{i\in[k]}\frac{1}{a_i}.$$

Sparsest-obstruction conjecture. If

$$\mathrm D(A)\leq 1-\frac{a_1-1}{a_1(a_1+1)},$$

then $A$ is schedulable. Equivalently, the instances $(2,3,a_3)$ and $(3,4,4,a_4)$, together with the indicated higher-density obstruction family for $a_1>3$, are conjectured to be the sparsest unschedulable instances beginning with each fixed value of $a_1$. This conjecture arose from the paper's computer experiment and is presented as requiring a better understanding of the proof; no resolution is given.