Maximal ideals in the Hecke algebra at $e=2$ in positive characteristic

Prove that, for a Hecke algebra $H_\infty$ over a field $F$ of positive characteristic with quantum parameter of quantum characteristic $e=2$, there are precisely two maximal ideals.

Background

The paper proves that the infinite Hecke algebra $H_\infty$ has precisely $e-1$ maximal ideals when $e>2$, and proves that it has exactly one maximal ideal in characteristic zero when $e=2$. For positive characteristic and $e=2$, the methods used for the symmetric group algebra do not directly extend because they rely on its Hopf algebra structure.

The paper notes that at least two maximal ideals are known in this remaining case and formulates the precise conjecture that there are no others. A proof would complete the analogous maximal-ideal classification for the relevant Hecke algebras.