De Pierro's conjecture on underrelaxed projection limits

Let $X$ be a real Hilbert space and let $C_1,\ldots,C_N$ be nonempty closed convex subsets of $X$, with metric projections $P_{C_i}$. For $\lambda\in]0,1]$, define

$$Q_{\lambda}:=((1-\lambda)Id+\lambda P_{C_N})\cdots((1-\lambda)Id+\lambda P_{C_2})((1-\lambda)Id+\lambda P_{C_1}).$$

Assume that $F_{\lambda}:=\operatorname*{Fix}Q_{\lambda}\neq\emptyset$ for every $\lambda\in]0,1]$. For $x\in X$, let

$$x_{\lambda}=\operatorname*{weak}\lim_{n\rightarrow+\infty}Q_{\lambda}^{n}(x).$$

Let $\mathcal{L}$ be the set of least-squares solutions,

$$\mathcal{L}:=\left\{x\in X\mid \sum_{i=1}^{N}\left\Vert x-P_{C_i}(x)\right\Vert^2=\inf_{y\in X}\sum_{i=1}^{N}\left\Vert y-P_{C_i}(y)\right\Vert^2\right\}.$$

De Pierro's conjecture. The weak limits satisfy

$$\lim_{\lambda\rightarrow0^{+}}x_{\lambda}=P_{\mathcal{L}}(x).$$

The conjecture concerns the behavior, as the underrelaxation parameter tends to zero, of weak limits generated by cyclic compositions of underrelaxed projections for inconsistent convex feasibility problems. It is false in general: a system of three compact convex sets in $\mathbb{R}^{3}$ provides a counterexample.