Sign conjecture for the $SL_2(\mathbb{R})$-Casson invariant of Brieskorn homology spheres

Let $\Sigma(m,p,q)$ be the Brieskorn homology sphere considered above, with $g$ the relevant genus, and let $R^{\rm zd}(\Sigma(m,p,q))=R^{\rm zd}(W_1)\cap R^{\rm zd}(W_2)$ denote the finite set of non-trivial Zariski-dense representations $f:\pi_1(\Sigma(m,p,q))\to SL_2(\mathbb{R})$. For such a representation, let $\tau^0_f$ be its Reidemeister torsion and write

$$(-1)^g\operatorname{sign}(\tau^0_f)=\varepsilon_f\in\{\pm1\}.$$

Sign conjecture. For every $f\in R^{\rm zd}(\Sigma(m,p,q))$, the quantity $\varepsilon_f$ is negative. Consequently,

$$\lambda_{SL_2(\mathbb{R})}(\Sigma(m,p,q))=-\lvert R^{\rm zd}(\Sigma(m,p,q))\rvert.$$

The authors report verification by computer when $m,p,q\leq 9$ or when $(m,p,q)=(m,2,3)$ with $m<25$; the conjecture predicts the sign for every representation in the finite representation set and determines the associated $SL_2(\mathbb{R})$-Casson invariant accordingly.