Characterize when propensity-weight derivatives determine graphical versus law-dependent optimality

Determine whether, for weighted average treatment effects with smooth propensity-dependent weights, the distinction between graph-only and law-dependent optimal adjustment-set optimality is governed by the derivative of the weight function with respect to the propensity score.

Background

The paper proves that constant weights, including the ATE weight, prevent outcome-side supplementation from increasing the efficiency bound, whereas every smooth positive weight with a nonzero propensity derivative admits a construction in which supplementation increases the bound. It also proves that ATT-optimal adjustment sets need not be determined by the graph alone, while the ATE has a graph-only optimal set.

These results motivate, but do not establish, a broader characterization linking the derivative of the propensity-dependent weight to whether optimal adjustment can be determined solely from the causal graph or instead depends on the underlying probability law. The question is explicitly posed as a conjecture and remains unresolved.