Stanley–Stembridge decomposition conjecture for immanant characters

Let $\gamma^\theta_{\mu/\nu}$ be the immanant character associated with a partition $\theta \vdash N$ and a skew shape $\mu/\nu$ of size $N$. In the case $\theta=(N)$, call $\Gamma^{(N)}_{\mu/\nu}$ the Stanley–Stembridge character.

Stanley–Stembridge decomposition conjecture. Every immanant character $\Gamma^\theta_{\mu/\nu}$ is a non-negative integral sum of Stanley–Stembridge characters.

This conjecture would reduce the general immanant-character conjecture to the Stanley–Stembridge conjecture, whose special case corresponds to $\theta=(N)$. It therefore proposes a structural decomposition of all immanant characters into the better-known Stanley–Stembridge characters.