Trust-region radius convergence conjecture for probabilistically fully quadratic models

Let $\{M_k\}$ be a model sequence that is probabilistically $(\kappa_{eh},\kappa_{eg},\kappa_{ef})$-fully quadratic for positive constants $\kappa_{eh}$, $\kappa_{eg}$, and $\kappa_{ef}$. Let $\{X_k\}$ be the sequence of random iterates generated by Algorithm~, and let $\{\tau_k\}$ denote its trust-region radii.

Trust-region radius convergence conjecture. Almost surely,

$$\lim_{k\to\infty}\tau_k=0.$$

The claim concerns whether the trust-region radius must shrink to zero for the probabilistic trust-region framework. The surrounding discussion explains that the analogous second-order stationarity result for deterministic fully quadratic models had not been extended to probabilistically fully quadratic models because successful iterations, and hence radius increases, are not guaranteed.