Complexity of 2D Snake Cube Puzzles

Given a chain of $HW$ cubes where each cube is marked "turn 90^∘" or "go straight", when can it fold into a 1 × H × W rectangular box? We prove several variants of this (still) open problem NP-hard: (1) allowing some cubes to be wildcard (can turn or go straight); (2) allowing a larger box with empty spaces (simplifying a proof from CCCG 2022); (3) growing the box (and the number of cubes) to 2 × H × W (improving a prior 3D result from height $8$ to $2$); (4) with hexagonal prisms rather than cubes, each specified as going straight, turning 60^∘, or turning 120^∘; and (5) allowing the cubes to be encoded implicitly to compress exponentially large repetitions.