Haines's invariance conjecture under isomorphisms

Let $bbalpha:bbG\tobbG'$ be an isomorphism of connected reductive groups, let $bbpi\inbbPi_{bbC}(bbG)$, and let ${}^{bbalpha}bbpi$ be the irreducible representation of $bbG'$ obtained by pre-composing with $bbalpha^{-1}$. Haines's invariance conjecture. The induced isomorphism ${}^Lbbalpha:{}^LbbG'(bbC)\to{}^LbbG(bbC)$ takes the $bbhat{bbG'}(bbC)$-conjugacy class of $(bbmathcal{LL}_{bbG'}({}^{bbalpha}bbpi))_{ss}$ to the $bbhat{bbG}(bbC)$-conjugacy class of $(bbmathcal{LL}_{bbG}(bbpi))_{ss}$.

This is the semisimplified form of Haines's invariance conjecture under isomorphisms. The supplied text gives no resolution status.