Conjecture for the extended irregular domination number of paths

Let $P_n$ be the path on $n$ vertices, and let $\gamma_e(P_n)$ denote its extended irregular domination number. Path-domination conjecture. For $n\geq 4$,

$$\gamma_e(P_n)=\begin{cases}\left\lceil\frac{n+1}{2}\right\rceil&\text{if }n=6,10,\left\lceil\frac{n+3}{2}\right\rceil&\text{otherwise.}\end{cases}$$

The equality is known for $n\in[4,26]\setminus\{6,10\}$, while the source establishes the exceptional lower value at $n=6,10$; the asserted formula for all $n\geq 4$ remains open.