Saddle-point avoidance conjecture for backtracking gradient descent

Let $f:\mathbb{R}^m\rightarrow\mathbb{R}$ be a $C^1$ function that is $C^2$ near its critical points, and let $\{z_n\}$ be the sequence generated by the Backtracking GD method from an initial point $z_0\in\mathbb{R}^m$. A saddle-point avoidance conjecture asserts that the set of initial points $z_0\in\mathbb{R}^m$ for which the cluster points of $\{z_n\}$ contain a saddle point has Lebesgue measure $0$. The preceding discussion presents this as a heuristic expectation based on the conjectured long-run stabilization of Backtracking GD to a finite union of Standard GD processes; whether saddle-point avoidance holds under the stated $C^1$ and local $C^2$ assumptions remains open.