The critical gradient extrapolation conjecture

Consider gradient descent run for a fixed number $N$ of steps with constant stepsize $h\in(0,1]$, starting from $x_0$ with $\lVert x_0-x_\star\rVert=D$. For simple extrapolation by a factor $c$, let $x_\sigma$ denote the extrapolated point and let $L$ be the smoothness constant. Critical gradient extrapolation conjecture. There exists $c_{\mathrm{crit,grad}}=1+\Theta(1/N)$ such that, for every $c\in[1,c_{\mathrm{crit,grad}}]$, simple extrapolation has the tight worst-case bound

$$\lVert\nabla f(x_\sigma)\rVert\leq\frac{LD}{Nhc+1}.$$

The conjecture formalizes the observed $1+O(1/N)$ shrinkage of the critical extrapolation factor for gradient-norm performance and predicts that the simple bound remains tight throughout this interval.