Conjecture on gradient-descent performance in overparameterized neural networks

Suppose a neural net has $O(n)$ parameters and the input data are generic. Let $F$ be its loss, and define the gradient-descent algorithm by

$$\mathcal{A}(\theta)=\theta-\eta\nabla F(\theta).$$

For an initial point $\theta$ drawn randomly from a specified distribution, define

$$\phi_{\mathcal{A}}(\theta)=\liminf_{k\rightarrow\infty}F(\mathcal{A}^k(\theta)).$$

Gradient-descent performance conjecture. For a proper constant $\eta$, and for a random initial point drawn from a certain distribution, such as Xavier initialization,

$$\Pr\bigl(\phi_{\mathcal{A}}(\theta)<\epsilon\bigr)>1-\delta,$$

where $\epsilon$ and $\delta$ are certain small constants. This informal conjecture proposes high-probability near-zero limiting loss for gradient descent in an overparameterized, generic-data regime; the source does not provide a proof or precise conditions for the constants and initialization distribution.