An Amir-Cambern theorem for subspaces of Banach lattice-valued continuous functions

For $i=1,2$, let $E_i$ be a reflexive Banach lattice over $\mathbb{R}$ with a certain parameter $\lambda^+(E_i)>1$, let $K_i$ be a locally compact (Hausdorff) topological space and let $\mathcal{H}_i$ be a closed subspace of $\mathcal{C}_0(K_i, E_i)$ such that each point of the Choquet boundary $\mathcal{Ch}_{\mathcal{H}_i} K_i$ of $\mathcal{H}_i$ is a weak peak point. We show that if there exists an isomorphism $T\colon \mathcal{H}_1 \to \mathcal{H}_2$ with $\Vert T \Vert \cdot \Vert T^{-1} \Vert<\min \lbrace \lambda^+(E_1), \lambda^+(E_2) \rbrace$ such that $T$ and $T^{-1}$ preserve positivity, then $\mathcal{Ch}_{\mathcal{H}_1} K_1$ is homeomorphic to $\mathcal{Ch}_{\mathcal{H}_2} K_2$.