The block-isomorphism character-triple conjecture for finite reductive groups

Let ${\mathbf{G}}$ be a finite reductive group with Frobenius map $F$, let $\ell$ be a prime, and let $e$ be the associated parameter. For every $\ell$-block $B$ of ${\mathbf{G}}^F$ and $d\geq 0$, let ${\mathcal{L}}^d(B)_\pm/{\mathbf{G}}^F$ be the sets of ${\mathbf{G}}^F$-orbits of the relevant even and odd local quadruples, with each orbit determining pairs $(\sigma,\vartheta)$ and their associated character triples. Put $X:={\mathbf{G}}^F\rtimes {\rm Aut}_{\mathbb F}({\mathbf{G}}^F)$.

The block-isomorphism character-triple conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$ and $d\geq 0$, there is an ${\rm Aut}_{\mathbb F}({\mathbf{G}}^F)_B$-equivariant bijection

$$\Lambda:{\mathcal{L}}^d(B)_+/{\mathbf{G}}^F\to{\mathcal{L}}^d(B)_-/{\mathbf{G}}^F$$

such that the character triples associated with corresponding local pairs are ${\mathbf{G}}^F$-block isomorphic in the sense of Späth's definition. This is intended as a structural counterpart to the numerical local-global counting conjectures, but the source gives no resolution evidence.