The exact L(3,2,1)-labeling number of a 4-valent circulant with steps 1 and 3

Let $n\in \mathbb{N}$ be even with $n\geq 142$, and let $G=C_n(\\{1,3,n-3,n-1\\})$. The invariant $\lambda_{(3,2,1)}(G)$ is the minimum span of an $L(3,2,1)$-labeling of $G$, meaning a labeling of the vertices by nonnegative integers such that labels of vertices at distances $1$, $2$, and $3$ differ by at least $3$, $2$, and $1$, respectively.

Exact labeling-number conjecture.

$$\lambda_{(3,2,1)}(G)= \begin{cases} 11 & \text{if }12\mid n,\\\\ 12 & \text{otherwise.} \end{cases}$$

The preceding theorem establishes the upper bound $\lambda_{(3,2,1)}(G)\leq 12$ for every even $n\geq142$, while the conjectured exact values require the corresponding lower bounds; the statement is presented as a conjecture in the source.