ACM tilting bundles on a Geigle-Lenzing projective plane of type $(2,2,2,p)$

Let $\mathbb{X}$ be a Geigle-Lenzing projective plane of type $(2,2,2,p)$ and $\mathsf{c}\mathsf{o}\mathsf{h}\mathbb{X}$ the category of coherent sheaves on $\mathbb{X}$. This paper is devoted to study ACM tilting bundles over $\mathbb{X}$, that is, tilting objects in the derived category ${\mathsf{D}}^{\mathrm{b}}(\mathsf{c}\mathsf{o}\mathsf{h}\mathbb{X})$ that are also ACM bundles. We show that a tilting bundle consisting of line bundles is the $2$-canonical tilting bundle up to degree shift. We also provide a program to construct ACM tilting bundles, which give a rich source of (almost) $2$-representation infinite algebras. As an application, we give a classification result of ACM tilting bundles.