In previous work of this author it was conjectured that the sum of power sums $p_λ,$ for partitions $λ$ ranging over an interval $[(1^n), μ]$ in reverse lexicographic order, is Schur-positive. Here we investigate this conjecture and establish its truth in the following special cases: for $μ\in [(n-4,1^4), (n)]$ or $μ\in [(1^n), (3,1^{n-3})], $ or $μ=(3, 2^k, 1^r)$ when $k\geq 1$ and $0\leq r\leq 2.$ Many new Schur positivity questions are presented.