Exact convergence-rate conjecture for Douglas–Rachford splitting on smooth convex composites

Consider the convex composite problem

$$\min_x f(x)+g(x),$$

where $f$ and $g$ are closed proper convex functions. Assume that the Douglas–Rachford operator has a fixed point $w^\star$, that $f$ is $L$-smooth, and write $x^\star=\operatorname{prox}_{\gamma f}(w^\star)$. Let $\{(x^k,y^k,w^k)\}$ be generated by the Douglas–Rachford splitting algorithm with stepsize $\gamma\in\left(0,\frac{2\sqrt{2}-1}{L}\right)$, relaxation parameter $\lambda\in\left(0,\frac{1+\sqrt{5}}{2}\right)$, and initial point $w^1$. Exact convergence-rate conjecture.

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

This is proposed as the exact objective-gap rate for Douglas–Rachford splitting in the smooth convex composite setting, based on numerical experiments in the performance-estimation framework; its resolution is not given.