Modal logics of Minkowski spacetime

For each $n$, let \mathcal{F}^n = \langle \mathbb{R}^{1+n}, \mathrel{\scalebox{0.85}{\oldalpha}} \rangle be the frame of $(1+n)$-dimensional Minkowski spacetime with causal relation \mathrel{\scalebox{0.85}{\oldalpha}}. Minkowski-spacetime logic conjecture.

1. $\mathcal{L}(\mathcal{F}^2) = \mathbf{D4da_2.2}$.
2. $\mathcal{L}(\mathcal{F}^{n}) = \mathbf{D4da.2}$ for $n \geq 2$.

These equalities propose complete axiomatizations of the normal modal logics determined by two-dimensional and higher-dimensional Minkowski spacetime, respectively. The surrounding discussion develops the relevant axioms and separates the two-dimensional case from higher-dimensional settings, but does not establish either equality.