  Let $\alpha :1,(1,\sqrt{x},\sqrt{y})^{\wedge }$ be a weight sequence with
Stampfli's subnormal completion and let $W_{\alpha }$ be its associated
weighted shift. In this paper we discuss some properties of the region
$\mathcal{U}:\mathcal{=}\{(x,y):W_{\alpha }$ is semi-cubically hyponormal$\}$
and describe the shape of the boundary of $\mathcal{U}$. In particular, we
improve the results of \cite[Theorem 4.2]{LLB} with properties of
$\mathcal{U}$.
