Computational complexity of Freeze-Tag in the plane for norms other than ℓ2

Ascertain the computational complexity of the Freeze-Tag Problem in (ℝ^2, ℓp) for norms other than ℓ2, including determining whether the problem is NP-hard in (ℝ^2, ℓ1).

Background

It is known that Freeze-Tag is NP-hard in three dimensions for all ℓp norms and is also NP-hard in two dimensions under the ℓ2 norm. However, the status for other planar norms, notably ℓ1, has not been settled.

Some prior work suggests NP-hardness in (ℝ2, ℓ1) may hold, but formal proofs remain absent; resolving this would complete the complexity landscape for the planar geometric variants across norms.

References

Similarly, FTP is NP-hard for $(\mathbb{R}2, \ell_2)$, though complexity results for other norms remain unresolved. It is suspected that FTP is also NP-hard for $(\mathbb{R}2, \ell_1)$.

— Improved Wake-Up Time For Euclidean Freeze-Tag Problem (2507.16269 - Alipour et al., 22 Jul 2025) in Related Work subsection (Introduction)