Fourier–Mukai wall-avoidance conjecture for rank-one Donaldson–Thomas classes

Let $(\beta,n)\in\mathbb{Z}^2$ satisfy

$$(1,0,-\beta,-n)=\gamma_1+\gamma_2,\qquad \gamma_i\in\mathcal{C},\qquad \Theta(\gamma_1)<\Theta(\gamma_2).$$

Assume $\beta\neq 0$ or $n>0$. Fourier–Mukai wall-avoidance conjecture. For every integer solution $(c,d)$ of

$$d^3-3\beta c^2d-nc^3=1,$$

one has

$$-\frac{d}{c}\notin(\Theta(\gamma_1),\Theta(\gamma_2)).$$

This is a constraint on possible Fourier–Mukai transforms and destabilizing decompositions in the study of Donaldson–Thomas invariants; the source presents it as a question/conjectural assertion and supplies no resolution in the excerpt.