Deré's real definability conjecture for complex Lie algebras

Let $L$ be a finite-dimensional complex Lie algebra, and let its complex conjugate be the complex Lie algebra obtained by applying complex conjugation to the structure constants of $L$. A complex Lie algebra $L$ is defined over $\mathbb{R}$ if there is a finite-dimensional real Lie algebra $L_{\mathbb{R}}$ such that

$$L\simeq L_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}.$$

Deré's conjecture. If $L$ is isomorphic to its complex conjugate, then $L$ is defined over $\mathbb{R}$; equivalently, there exists a real Lie algebra $L_{\mathbb{R}}$ such that

$$L\simeq L_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}.$$

The conjecture is false: Cyril Demarche disproved it, and the paper constructs an explicit 10-dimensional complex two-step nilpotent Lie algebra that is isomorphic to its complex conjugate but is not definable over $\mathbb{R}$.