Real-part conjecture for Ricci-negative derivations

Let $\mathfrak{n}$ be a nilpotent Lie algebra and let $\operatorname{Der}(\mathfrak{n})_{\mathrm{rn}}$ denote the derivations whose associated solvable extension admits a Ricci-negative metric. For $D\in\operatorname{Der}(\mathfrak{n})$, let $D^{\mathbb{R}}$ be the real semisimple part in its additive Jordan decomposition. Real-part conjecture.

$$D\in\operatorname{Der}(\mathfrak{n})_{\mathrm{rn}}\quad\Longleftrightarrow\quad D^{\mathbb{R}}\in\operatorname{Der}(\mathfrak{n})_{\mathrm{rn}}.$$

If true, this would reduce the Ricci-negative derivation problem to real-diagonalizable derivations; the source explicitly presents it as a necessary consequence of the preceding characterization conjecture.