Cocoercive-subdifferential convergence-rate conjecture for Douglas–Rachford splitting

Let $A$ and $B$ be operators satisfying the paper's standing Assumption, let $B$ be $\beta$-cocoercive, and let $A=\partial g$ for a closed proper convex function $g$. Let $\{w^k\}$ be generated by the Douglas–Rachford splitting algorithm with stepsize $\gamma\in(0,\beta)$ and relaxation parameter $\lambda\in(0,2)$, and let $T$ be its Douglas–Rachford operator with fixed point $w^\star$. Cocoercive-subdifferential convergence-rate conjecture. The residual after $N$ iterations satisfies

$$\left\|T(w^N)-w^N\right\|^2\leq\frac{\lambda^2}{((N-1)\lambda+1)^2}\left\|w^1-w^\star\right\|^2.$$

The claim proposes that subdifferential structure of $A$, unlike cocoercivity of $B$ alone, permits an improved rate; it is presented as a conjecture informed by performance-estimation experiments, and no resolution is supplied in the source.