Whether separate flat and displacement monotonicity imply the joint flat/displacement condition

Determine whether the following implication holds for a function g: R^n × M_{2−r}(R^n) → R. Assume that g satisfies both (i) flat non-increasing (anti-Lasry–Lions) monotonicity: for all finite nonnegative measures m,m′ ∈ M_{2−r}(R^n), ∫ (g(x,m) − g(x,m′)) d(m − m′)(x) ≤ 0, and (ii) displacement non-decreasing monotonicity: for all R^n-valued random variables X and X′ with suitable integrability, E[(∇_x g(X, P_X) − ∇_x g(X′, P_{X′})) · (X − X′)] ≥ 0. Ascertain whether these two properties together imply the joint flat non-increasing/displacement non-decreasing inequality: for any nonnegative random variables q,q′ with E[q],E[q′] ≤ exp(α T) and any R^n-valued random variables X,X′ with E[(q+q′)(|X|^{2−r}+|X′|^{2−r})] < ∞, E[(q − q′)(g(X,(qP)_X) − g(X′,(q′P)_{X′}))] − E[(q ∇_x g(X,(qP)_X) − q′ ∇_x g(X′,(q′P)_{X′})) · (X − X′)] ≤ 0.

Background

In Subsection 4.4, the authors introduce a uniqueness criterion for equilibria based on a joint monotonicity property of the mean-field interaction g, termed flat non-increasing/displacement non-decreasing. This joint property is expressed by an inequality involving both the measure-dependence of g (through the effective measures (qP)X and (q′P){X′}) and the gradient in x, and it plays a central role in proving uniqueness of robust mean-field game equilibria.

They show that this joint condition implies two familiar separate conditions: displacement monotonicity (by taking q=q′=1) and a flat, anti-Lasry–Lions monotonicity condition in the measure argument (by taking X=X′). However, it is unclear whether these two separate properties suffice to recover the full joint inequality, which would simplify verification of uniqueness in applications. The remark explicitly highlights this gap.