De Pierro's least-squares convergence conjecture for under-relaxed cyclic projections

Let $C_1,\ldots,C_m$ be nonempty closed convex subsets of a Hilbert space $\mathcal{H}$. For a fixed starting point $u_0$, let $\mathbf{u^\varepsilon}$ denote the corresponding limit cycle of the $\varepsilon$-under-relaxed cyclic projection method, and let $\mathbf{v}_k$ be the modified process generated using a sequence $\lambda_k\to 0$ with $\sum_k\lambda_k=+\infty$. Define

$$S=\operatorname*{\arg\,\min}_{u\in\mathcal{H}}\sum_{i=1}^m d(u,C_i)^2.$$

De Pierro's conjecture. The least-squares solution set $S$ is nonempty if and only if, for every starting point $u_0$, the limits

$$\mathbf{u}=\lim_{\varepsilon\downarrow 0}\mathbf{u^\varepsilon},\qquad \mathbf{v}=\lim_{k\to\infty}\mathbf{v}_k$$

exist, with both diagonal representatives $\bar u$ and $\bar v$ belonging to $S$.

The conjecture connects convergence of under-relaxed cyclic-projection cycles and the associated modified process with solvability of the least-squares problem. It is known under various conditions, but the paper gives a counterexample in general: three compact convex sets in $\mathbb{R}^3$ for which the $\varepsilon$-under-relaxed cycles do not converge.