Cyclic Douglas–Rachford best approximation conjecture

Let $\mathcal H$ be the underlying Hilbert space, and let $C_1,C_2\subseteq\mathcal H$ be closed and convex with $C_1\cap C_2=\emptyset$. Write $P_{C_i}$ for the metric projection onto $C_i$. A pair $(u,v)\in C_1\times C_2$ is a best approximation pair when $\|u-v\|=\inf\{\|c_1-c_2\|:c_1\in C_1,\ c_2\in C_2\}$. The two-set cyclic Douglas–Rachford scheme is the iteration generated by the corresponding cyclic Douglas–Rachford operator.

Cyclic Douglas–Rachford best approximation conjecture. If a best approximation pair relative to $(C_1,C_2)$ exists, then the two-set cyclic Douglas–Rachford scheme converges weakly to a point $x$ such that $(P_{C_1}x,P_{C_2}x)$ is a best approximation pair relative to $(C_1,C_2)$.

The conjecture concerns convergence in the inconsistent case, where the two closed convex sets are disjoint. The supplied source explicitly notes that non-convex settings can make the conjecture false; for the stated closed convex setting, the claim is presented as a conjecture and is therefore recorded here as refuted only according to the supplied status evidence.