The perfect extremal Skolem set conjecture

Let $A=\{a_1,a_2,\dots,a_n\}$ be a set with $a_1>a_2>\dots>a_n$ and total sum $\sum_{i=1}^n a_i=n^2$. A perfect extremal Skolem set is a perfect Skolem set satisfying this extremal equality. The perfect extremal Skolem set conjecture. The set $A$ is a perfect extremal Skolem set if and only if the number of even $a_i$'s is even and

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

for each $1\leq m\leq n$. This is the extremal special case of the perfect Skolem set conjecture. The source notes that the analogous assertion fails for multisets, but gives no separate resolution for ordinary sets.