On a sequence of singular ball quotient surfaces on the line $K^2=9\chi-18$

Starting from computer experiments with the fundamental group of the Cartwright--Steger surface, we construct an infinite tower $(X_n)_{n\ge 1}$ of normal projective surfaces obtained by successive $\mathbb Z/3$-Galois covers $X_{n}\to X_{n-1}$. For $n>1$, their minimal resolutions $\widetilde{X}_n$ lie on the line $K^2 = 9\chi - 18$ (equivalently $c_1^2 = 3c_2 - 72$), which is parallel to the Bogomolov--Miyaoka--Yau line $K^2 = 9\chi$ of ball quotients. We compute the fundamental groups for the first cases, showing that $\pi_1(\widetilde{X}_n)=1$ for $n=1,\ldots,5$. Motivated by the geometry of the construction, we conjecture that all $\widetilde{X}_n$ are simply connected.