Varieties of subalgebras and endotrivial modules

Let $(\mathfrak{g},[p])$ be a finite dimensional restricted Lie algebra over a perfect field $\text{mathbbm}k$ of characteristic $p\text{\hspace{0.17em}⁣}\ge \text{\hspace{0.17em}⁣}3$. By combining methods from recent work of Benson-Carlson \cite{BC20} with those of \cite{CF21,Fa17} we obtain a description of the endotrivial $(\mathfrak{g},[p])$-modules in case $\mathfrak{g}$ is supersolvable.