NP-hardness of 3-uniform linearly-ordered colouring

Prove that $PCSP(\LO^3_k,\LO^3_\ell)$ is NP-hard for every pair of constants 2 ≤ k ≤ ℓ, including the unresolved case $\PCSP(\LO_2^3,\LO_3^3)$.

Background

Linearly-ordered, or unique-maximum, hypergraph colouring requires the maximum colour in every hyperedge to occur uniquely. The paper establishes NP-hardness for all constant 3 ≤ k ≤ ℓ and uniformity r ≥ 4, but does not resolve the 3-uniform case when k = 2. The broader conjecture attributed to Barto et al. asserts NP-hardness for all constant 2 ≤ k ≤ ℓ and r = 3; the paper explicitly notes that even the case (k, ℓ) = (2, 3) remains unresolved.

References

Barto at al. conjectured that $\PCSP(\LO_k3,\LO_\ell3)$ is \NP-hard for all constant $2\leq k\leq \ell$, but even the case $\PCSP(\LO_23,\LO_33)$ is still open.

— Complexity of approximate conflict-free, linearly-ordered, and nonmonochromatic hypergraph colourings  (2501.12062 - Nakajima et al., 21 Jan 2025) in Section 1, paragraph “Linearly-ordered colourings”