QUE conjecture for quasimodes on compact hyperbolic surfaces

Let $M$ be a compact hyperbolic surface. For a spectral parameter $r$, an $\omega(r)$-quasimode is a function $\psi$ satisfying

\left\\|(\Delta+(\tfrac14+r^2))\psi\right\\|_2\leq r\omega(r)\left\\|\psi\right\\|_2.

A sequence of $o(1/\log r)$-quasimodes has spectral-window width asymptotically smaller than $1/\log r$.

QUE conjecture for quasimodes. Any sequence of $o(1/\log r)$-quasimodes satisfies QUE.

The paper identifies $1/\log r$ as the conjectural threshold for QUE of quasimodes, in contrast with larger windows where equidistribution phenomena can persist even though QUE may fail on compact surfaces. The conjecture is unresolved in the supplied text.