Alt-GDA convergence lower-bound conjecture for non-quadratic objectives

Let ${\mathcal F}(\mu_x,\mu_y,L_x,L_y,L_{xy})$ denote the function class considered for the alternating gradient descent-ascent method, and let $\kappa_x$, $\kappa_y$, and $\kappa_{xy}$ be the corresponding condition numbers. For constant step sizes $\alpha,\beta>0$, consider the convergence complexity of Alt-GDA on a function $f$ in this class. Alt-GDA convergence lower-bound conjecture. There exists a non-quadratic function

$$f\in{\mathcal F}(\mu_x,\mu_y,L_x,L_y,L_{xy})$$

such that, for any constant step sizes $\alpha,\beta>0$, convergence of Alt-GDA requires iteration complexity

$$\Theta\left(\left(\kappa_x+\kappa_y+\kappa_{xy}(\kappa_x+\kappa_y)^p\right)\log\frac{1}{\epsilon}\right)$$

for $p\in(0,\tfrac12)$. This conjecture concerns whether the presently established convergence upper bound for Alt-GDA can be complemented by a matching lower-bound construction for a non-quadratic objective.