Asymptotic square-root conjecture for the optimal shrinkage parameter

Let $m$ denote the number of areas, let $\boldsymbol{\theta}_{( ,)}$ be the ordered random effects, and let $\boldsymbol{\theta}^{[2]}_{( ,)}(\boldsymbol{ ,})$ denote the predictor indexed by the shrinkage parameter $,\, with optimal choice$ ^o$minimizing its expected loss. Let$ ^*$be the corresponding variance ratio. **Asymptotic square-root conjecture.** The optimal$ , in the sense of Theorem 4, satisfies

$$\lim_{m\rightarrow\infty} ^o=\sqrt{ ^*}.$$

The preceding theorem only gives the range $[ ^*,\sqrt{ ^*}]$ asymptotically, while this conjecture predicts convergence to its upper endpoint. The source provides simulations but no proof or resolution.