The non-degeneracy conjecture for solutions of the non-critical $SU(3)$ Toda system

Let $n_1,n_2$ be the parameters of the Toda system $(\mathrm{Toda})$, with $n_1\not\equiv n_2\mod 3$ and $n_1<n_2$. A solution is non-degenerate when its linearized problem has only the trivial solution. Let $\tau$ be the torus period, with periods identified under the action of $SL(2,\mathbb{Z})$ by

A\mathbin{\cdot}\tau=\frac{a\tau+b}{c\tau+d},\qquad A=\begin{pmatrix}a&b\\\c&d\end{pmatrix}\in SL(2,\mathbb{Z}).

Non-degeneracy conjecture. Except for finitely many $\tau$ modulo the $SL(2,\mathbb{Z})$ action, every solution of the Toda system is non-degenerate. If true, this would support the generic exact solution count conjectured immediately before it; the supplied text gives no resolution.