On the stable Harbourne conjecture for ideals defining space monomial curves

For the ideal $\mathfrak{p}$ in $k[x, y, z]$ defining a space monomial curve, we show that $\mathfrak{p}^{(2 n - 1)} \subseteq \mathfrak{m} \mathfrak{p}^{n}$ for some positive integer $n$, where $\mathfrak{m}$ is maximal $(x, z)$. Moreover, smallest such $n$ determined. It turns out there counterexample to claim due Grifo, Huneke, and Mukundan, which states $\mathfrak{p}^{(3)} \mathfrak{p}^2$ if $k$ field of characteristic not $3$; however, stable Harbourne conjecture holds curves as they claimed.