Neighborhood-prime conjecture for unions of cycles with one arbitrary component

For positive integers $k_1,\ldots,k_\ell$ and a positive integer $n$, let $C_m$ denote the cycle graph of length $m$, and let $\cup$ denote disjoint union.

Cycle-union conjecture. The union

$$C_{4k_1}\cup C_{4k_2}\cup\cdots\cup C_{4k_\ell}\cup C_n$$

is neighborhood-prime if $n\equiv 0\pmod{4}$ or if $n$ is odd.

The conjecture is presented as a consequence that would follow from an affirmative solution of the Deretsky–Lee–Miller prime-labeling conjecture, together with the paper's neighborhood-graph theorem. The source does not report a resolution.