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