Recall the power-sum expansion of Schur functions, $$ s_\lambda = \sum_\mu \chi^{\lambda}(\mu) \frac{p_\mu}{z_\mu}, $$ in terms of Sn-character. These can be calculated by the Murnaghan-Nakayama rule, stating that characters are a signed sum over border-strip tableaux: $$ \chi^{\lambda}(\mu) = \sum_{T \in BST(\lambda,\mu)} (-1)^{ht(T)} . $$
Now, what if we instead consider the 'cancellation-free' expression (yes, the omission of $z_\mu$ is intended): $$ CF_\lambda := \sum_\mu |BST(\lambda,\mu)| p_\mu. $$
Is the symmetric function $CF_\lambda$ also Schur-positive? I have verified that this is the case for all $|\lambda| \leq 8$.
This is the data for all $\lambda$ with at most size 4.
1->ss[{1}]
2->2 ss[{2}]
11->2 ss[{2}]
3->3 ss[{3}]+ss[{2,1}]+ss[{1,1,1}]
21->5 ss[{3}]+3 ss[{2,1}]+ss[{1,1,1}]
111->3 ss[{3}]+ss[{2,1}]+ss[{1,1,1}]
4->5 ss[{4}]+3 ss[{2,2}]+2 ss[{3,1}]+2 ss[{2,1,1}]+ss[{1,1,1,1}]
31->10 ss[{4}]+6 ss[{2,2}]+10 ss[{3,1}]+6 ss[{2,1,1}]+2 ss[{1,1,1,1}]
22->7 ss[{4}]+7 ss[{2,2}]+6 ss[{3,1}]+2 ss[{2,1,1}]+3 ss[{1,1,1,1}]
211->10 ss[{4}]+6 ss[{2,2}]+10 ss[{3,1}]+6 ss[{2,1,1}]+2 ss[{1,1,1,1}]
1111->5 ss[{4}]+3 ss[{2,2}]+2 ss[{3,1}]+2 ss[{2,1,1}]+ss[{1,1,1,1}]
