Whether horizon-dependent Heavy-Ball schedules achieve Nesterov’s O(T^-2) rate

For each horizon $T\ge 2$, consider the Heavy-Ball iteration $x_{t+1}=x_t-\eta_t\nabla f(x_t)+\beta_t(x_t-x_{t-1})$, with predetermined horizon-dependent parameters satisfying $\eta_t\ge 0$ and $0\le\beta_t<1$, and with zero initial velocity $x_{-1}=x_0$. Does there exist such a parameter schedule for which, for every convex $1$-smooth function $f$ and every initialization satisfying $\|x_0-x^\star\|\le 1$ for some minimizer $x^\star$, the last iterate satisfies $f(x_T)-f^\star=O(T^{-2})$? The cited lower-bound claim asserts that for every such schedule there is a convex $1$-smooth objective, with the stated initialization condition, for which $f(x_T)-f^\star=\Omega\!\left(\frac{1}{T^\alpha\log T}\right)$, where $\alpha=\frac{1+\sqrt{5}}{2}$, thereby ruling out the target rate for this class if the claim is correct.