Conjecture on interior equality of quenched and averaged rate functions

Consider nearest-neighbor random walk in a random environment on $\mathbb{Z}^d$ with an i.i.d. uniformly elliptic environment. Let $I_q$ and $I_a$ denote the quenched and averaged large-deviation rate functions, respectively. Interior equality conjecture. For walks in high dimension, $I_a$ and $I_q$ agree on a set with non-empty interior. The rate functions are already known to agree at the origin and on their zero sets in dimensions at least two, but it is not known whether these are the only points of equality; the conjecture predicts a substantially larger equality set in high dimension.