Variance inflation for multidimensional continuous feature maps

Prove or disprove that variance inflation relative to complete randomization persists for continuous covariate-adaptive randomization when the covariate feature map has dimension $p'\geq 2$.

Background

The paper derives an explicit worst-case variance-inflation result only for scalar continuous feature maps. It notes that the corresponding higher-dimensional analysis involves a vector Poisson equation, which is difficult to solve. Numerical evidence is reported, but a general theoretical result is not provided.

References

Whether the inflation persists with $p'\geq 2$ is not discussed by Proposition~\ref{prop:scalar-ccar}. In higher dimensions, the associated vector Poisson equation is hard to solve. We conjecture that similar variance inflation still persists, and report supporting numerical evidence in Example \ref{example:instability} in Appendix \ref{app:additional sim}.

— Discretization in covariate-adaptive randomization: gains and losses  (2609.11012 - Zhao et al., 10 Sep 2026) in Remark 2, “inflation conjecture,” following the scalar variance-inflation example