Three local $π$-characters

We answer in ZFC a question of N. Carlson by constructing a zero-dimensional Tychonoff space $X$, with exactly one non-isolated point, for which every point admits a decreasing local $π$-base and \[ πχ(X)=πχ_{\mathrm l}(X)=ω<πχ_{\mathrm c}(X)=ω_1 <πχ_{\mathrm d}(X)=ω_2. \] A preliminary chain-refine