The nilpotent characterization conjecture for $\Theta$-positive unipotents

Let $G$ be a semisimple Lie group with a $\Theta$-positive structure, with nilpotent Lie algebra $\mathfrak{u}_\Theta$ and positivity cones $c^\circ_\beta$ for $\beta\in\Theta$. The nilpotent characterization conjecture. If

$$v=\sum_{\beta\in\Theta}v_\beta,\qquad v_\beta\in c^\circ_\beta,$$

then $\exp(v)\in U_\Theta^{>0}$. Conversely, if $v\in\mathfrak{u}_\Theta$ and $\exp(tv)\in U_\Theta^{>0}$, then

$$v=\sum_{\beta\in\Theta}v_\beta,\qquad v_\beta\in c^\circ_\beta.$$

This conjecture is intended to characterize the positive unipotent elements through their nilpotent logarithms; the source gives no resolution.