Edge irregularity strength conjecture for cycle-star graphs

Let $G=CS_{k,n-k}$ be a cycle-star graph, where $k$ is the cycle order and $n-k$ is the number of leaves. The edge irregularity strength $es(G)$ is the least positive integer $s$ for which there is a vertex labeling $\phi:V(G)\to\{1,2,\ldots,s\}$ such that the sums of labels on distinct edges are distinct.

Cycle-star edge irregularity strength conjecture. For $k\geq 8$ and $n-k\geq 1$,

$$es(G)=\begin{cases} \left\lceil\dfrac{n+1}{2}\right\rceil & \text{for } k+1\leq n\leq 2k-4,\\ n-k+2 & \text{for } n\geq 2k-3. \end{cases}$$

This conjecture extends the values established in the paper for smaller cycle lengths, including the cases $k=7$. Its validity for all $k\geq 8$ is left open by the source.