Locally commuting operators and positivity on tensor product spaces

Question

I am reading this paper https://arxiv.org/abs/quant-ph/0501020 and two questions have been arisen for me: 1. In page 3 (left column) has been written: "Hence it follows that if $\tilde{S}_{k}^{(GHZ_{N})}$ and $\tilde{S}_{l}^{(GHZ_{N})}$ commute locally then the maximum of $\langle\tilde{S}_{k}^{(GHZ_{N})}+\tilde{S}_{l}^{(GHZ_{N})}\rangle$ for separable and entangled states coincide." But why? According to observation 1, these two operators have a common eigenstate, but why the expectation value in this common eigenstate is the maximum value necessarily? My second question is related to page 5 (left column): How can we prove $\mathcal{W}-2\widetilde{\mathcal{W}}^{GHZ_{N}}\geq 0$? For this, we need to prove $\langle\sum_{i}\bar{_{\alpha_{i}}}x_{i1}\otimes...\otimes x_{iN}|\mathcal{W}-2\widetilde{\mathcal{W}}^{GHZ_{N}}|\sum_{i}{\alpha_{i}}x_{i1}\otimes...\otimes x_{iN}\rangle\geq 0$ which is a very diffucult computation. Thank you!