Taketomi–Tamaru conjecture on congruent automorphism orbits and Ricci solitons

Let $\mathfrak{l}$ be an $n$-dimensional Lie algebra, and let $L$ be the connected and simply connected Lie group with Lie algebra $\mathfrak{l}$. The group $\mathbb{R}^{\times}\operatorname{Aut}(\mathfrak{l})$ acts on $O(n)\backslash GL_n(\mathbb{R})$ from the right. Taketomi–Tamaru's conjecture. If this action is not transitive, while all $\mathbb{R}^{\times}\operatorname{Aut}(\mathfrak{l})$-orbits are congruent to one another with respect to $GL_n(\mathbb{R})$, then $L$ does not admit left-invariant Ricci solitons. The claim was proposed by Taketomi and Tamaru and exhibited examples were given in which it holds; however, the stated conjecture is false, because Jablonski gave a counterexample.