Pirashvili's conjecture for finite-dimensional complex Lie algebras

Let $\mathfrak{g}$ be a non-trivial finite-dimensional complex Lie algebra. Say that $\mathfrak{g}$ satisfies the Pirashvili conditions if, equivalently, $HL_p(\mathfrak{g})=0$ for all $p\geq 1$, $HL^p(\mathfrak{g})=0$ for all $p\geq 1$, $H_p(\mathfrak{g},\mathfrak{g})=0$ for all $p\geq 0$, or $H^p(\mathfrak{g},\mathfrak{g})=0$ for all $p\geq 0$ together with $\mathfrak{g}$ perfect. Pirashvili's conjecture. The Lie algebra $\mathfrak{g}$ is semisimple if and only if it satisfies the Pirashvili conditions. The conditions imply that $\mathfrak{g}$ is perfect, complete, unimodular, sympathetic, and rigid; the conjecture asks whether they characterize semisimplicity among non-trivial finite-dimensional complex Lie algebras.