Nonexistence of spanning thin codes for prime modulus

Let $N$ be a prime with $N\geq 3$. A spanning simplex is a lattice simplex whose lattice points at height one generate the ambient lattice, and a non-degenerate thin code is a thin code that is not degenerate. Prime-modulus spanning-code conjecture. For prime $N\geq 3$ there are no non-degenerate thin codes corresponding to spanning simplices.

Computations in dimensions $d\leq 8$ suggest the conjecture, particularly for spanning simplices with prime $N_\Delta$. Its validity in general remains open.