The BNSR-invariant conjecture for the Stein groups $F_S^r$

Let $F_S^r$ be a Stein group indexed by a set $S$ and an integer $r$, let $\Sigma^m(F_S^r)$ denote its BNSR-invariant in degree $m$, and let $\Sigma(F_S^r)$ denote its character sphere. Write $[\lambda]$ and $[\rho]$ for the projective classes of the characters $\lambda$ and $\rho$.

BNSR-invariant conjecture. For any $S$ and $r$,

$$\Sigma^1(F_S^r)=\Sigma(F_S^r)\setminus\{[\lambda],[\rho]\},$$ $$\Sigma^2(F_S^r)=\Sigma(F_S^r)\setminus\{[a\lambda+b\rho]\mid a,b\ge 0\},$$

and

$$\Sigma^\infty(F_S^r)=\Sigma^2(F_S^r).$$

This conjecture predicts that the BNSR-invariants of the general Stein groups $F_S^r$ have the same pattern as those of $F_{2,3}$ and Thompson's group $F$. The preceding discussion emphasizes that even abelianizing general $F_S^r$ is technically difficult, so computing their BNSR-invariants remains open.