Existence of a universal hard instance for basic Newton’s method exceeding 1/ε^2 iterations

Determine whether there exists a single smooth function f:R^n→R such that the basic Newton’s method with unit steps (solving ∇^2 f(x^k) s^k = −∇ f(x^k) and taking x^{k+1} = x^k + s^k) requires more than 1/ε^2 iterations to find an iterate x with ∥∇ f(x)∥ ≤ ε for arbitrary ε > 0.

Background

For any prescribed tolerance ε1, an example can be constructed where basic Newton’s method needs more than 1/ε1^2 iterations to reach ∥∇f(x)∥ ≤ ε1, but the function in that construction depends on ε1.

It remains unresolved whether there exists a single function (independent of ε) that forces basic Newton’s method to require more than 1/ε^2 iterations for all sufficiently small ε.