The spectral equality conjecture for weighted composition operators on Banach lattice subspaces

Let $X$ be a Banach lattice with Stonean compact $K$, let $Y$ be the subspace and $T$ the weighted composition operator from Proposition 1, and let $S$ be the corresponding weighted composition operator on $C(\mathfrak{M})$, with boundary space $C(\partial)$. Assume the conditions of Proposition 1 and that the algebra $\mathcal{A}$ is almost localized in $C(K)$. The spectral equality conjecture.

$$\sigma(T,Y)=\sigma(S,C(\mathfrak{M})).$$

Moreover,

$$\sigma_{ap}(T,Y)=\sigma_{ap}(S,C(\partial)).$$

The preceding propositions establish the inclusion from the spectrum of $S$ into that of $T$, while examples show that equality can fail without an additional richness assumption on $\mathcal{A}$. The conjecture asserts that almost localization in $C(K)$ is sufficient for both the full-spectrum and approximate-point-spectrum equalities.