Monotonicity conjecture for algebraic stringy Euler numbers of Kawamata log-terminal pairs

Let $(X,\Delta)$ be a Kawamata log-terminal pair, and let $e^{\rm str}_{\rm alg}(X,\Delta)$ denote its algebraic stringy Euler number. Monotonicity conjecture. The algebraic stringy Euler number $e^{\rm str}_{\rm alg}(X,\Delta)$ is strictly monotone with respect to the elementary birational transformations from the Mori theory of Kawamata log-terminal pairs $(X,\Delta)$.

The source presents this as a generalization from varieties to Kawamata log-terminal pairs; it does not specify the direction of monotonicity for each transformation or give a resolution status.