Yes, I see ... the problem is that Octane doesn't full support UV mapping in the curves at point level, each curve only can have one UV map value for all the curve segments, so the UV map is useless in this object to map a texture on it.
The texture using the box projection in object space doesn't stick to the object because you are animating the object at SOP level. This is for Octane a deformation effect, so the texture applied in object space is not fixed to the geometry. If you animate the object at OBJ level (in this case your object translation), the texture is going to stick to the object without problems.
In sort: currently the Octane curves don't support UV mapping at point level, so you can't stick a texture in this kind of objects if you are animating deformations in the mesh (transformations at SOP level, for example). You can stick a texture using any projection in object space if you animate the object at OBJ level, animating the transformation parameters.
Computer Graphics Software Developer
Octane plugin development and support: LightWave and Houdini