Universal Smith normal form conjecture for the Adinkra Laplacian

Let $L$ be the Laplacian of an Adinkra, and let $\hat{L}$ be its polynomial lift over $\mathbb{Z}[x]$. Write the Smith normal form of $L$ as a tuple of invariant factors, with multiplicities indicated by exponents. Universal Smith normal form conjecture. If the Smith normal form of $L$ is

$$(1^{\#V/2},(N^2-N)^{\#V/2}),$$

then the Smith normal form of $\hat{L}$ exists over $\mathbb{Z}[x]$ and must be

$$(1^{\#V/2},(2(N-1)x+(N-1)(N-2))^{\#V/2}).$$

The paper explains that non-trivial invariant factors obstruct the existence of this universal Smith normal form and conjectures that the displayed converse holds; the claim remains open.