The perfect Skolem set conjecture

Let $A = \{a_1,a_2,\dots,a_n\}$ be a set of positive integers with $a_1 < a_2 < \dots < a_n$. A Skolem set is a set for which $\{1,\dots,2n\}$ can be partitioned into pairs whose differences are the elements of $A$. The perfect Skolem set conjecture. The set $A$ is a Skolem set if and only if the number of even $a_i$ is even and

$$\sum^n_{i=m} a_i \leq n^2-(m-1)^2$$

for each $1 \leq m \leq n$. This proposes that the elementary parity and density conditions are also sufficient for ordinary sets, although the corresponding existence problem for multisets is substantially harder.