Constrained T region-count conjecture

Prove the conjectured formula a_{\overline{\sf T}}(n) = 2n² + n + 1 − λn + 3\binom{λ}{2}, where λ = ceil(n/3), for the maximum number of regions formed by n constrained long-legged T shapes, and construct arrangements attaining the formula for every n ≥ 0.

Background

Two constrained long-legged T shapes consist of a line and a perpendicular ray meeting at a distinguished center, subject to similarities. Two copies intersect in at most four points, yielding the upper bound a_{\overline{T}}(n) ≤ 2n² + 1.

Computational experiments for n up to 12 suggest a correction term depending on λ = ceil(n/3). The authors explicitly state that, for n > 4, they have neither a construction achieving the proposed formula nor a proof of optimality, so the formula remains conjectural.