Silver-schedule convergence-rate conjecture for Douglas–Rachford splitting

In the convex composite problem with closed proper convex functions $f,g$, suppose the Douglas–Rachford operator has a fixed point $w^\star$ and $f$ is $L$-smooth. Let $\rho$ denote the silver-ratio constant and let $\pi_k$ denote the paper's silver stepsize-schedule parameter. Set the relaxation schedule to $\lambda=(\pi_k,1)$, with relaxation value $\lambda_i$ at iterate $i$, and let $N=2^k$. If $\{(x^k,y^k,w^k)\}$ is generated by the Douglas–Rachford algorithm with $\gamma\in\left(0,\frac{2\sqrt{2}-1}{L}\right)$, then Silver-schedule convergence-rate conjecture.

$$f(y^N)+g(y^N)-f(x^\star)-g(x^\star)\leq\frac{1}{4\gamma\rho^k}\left\|w^1-w^\star\right\|^2.$$

The conjecture predicts the rate obtained by tuning relaxation according to the silver schedule; the source motivates it through prior analyses and numerical evidence but does not establish it.