Suppose you are given a list of integer lengths, e.g., $(5,3,2,2,1,1,2,1,1)$. The task is to decide if they can form a closed cycle in $\mathbb{Z}^d$ by connecting segments of those lengths in order, each parallel to a coordinate axis, each connecting two lattice points, each orthogonal to its neighbors. This latter condition forbids connecting two segments collinearly: a $90^\circ$ turn is forced between each pair of segments.

In $\mathbb{Z}^2$, the decision problem is NP-complete,
because both the horizontal, and the vertical lengths must
partition equally to close: $5+1=2+2+2$ and $3+1=2+1+1$
in the example left below.

Thus given an even number of integers, interleaving 1's results in
a list that can form a cycle in $\mathbb{Z}^2$ iff the given integers
may be partitioned into two equal halves.
My question is:

Q. Is the decision problem NP-complete in $\mathbb{Z}^d$ for $d \ge 3$?

The choices of which direction to follow at each juncture seem to complicate a reduction from Partition. I feel like I am missing a simple argument here ... I'd appreciate it if anyone could supply it—Thanks!

**Addendum**.
A potentially more interesting question lay behind what I posed above: When might the list of lengths
be formed to realize the unknot in $\mathbb{Z}^3$?
I'll post that as a separate question if I can see how to formulate it sharply.