Endomorphism algebras of Gorenstein-projective modules over string algebras

Let $A$ be a string algebra. We identify its Gorenstein-projective support $\tau$-tilting pairs with a distinguished class of maximal good collections and give a finite compatibility graph for them. Under the G-condition, we construct an explicit labelled tiled-surface model for ${\mathrm{E}\mathrm{n}\mathrm{d}}_A(T)$ for every nonzero basic Gorenstein-projective module $T$. Under the same condition, we show that $A$ is representation-finite if and only if ${\text{End}}_A(T)$ is representation-finite for every basic Gorenstein-projective module $T$.