The Güntürk-Thao theorem revisited: polyhedral cones and limiting examples

In 2023, Güntürk and Thao proved that the sequence $(x^{(n)}{)}_{n\in \mathbb{N}}$ generated by random (relaxed) projections drawn from a finite collection of innately regular closed subspaces in a real Hilbert space satisfies ${\sum }_{n\in \mathbb{N}}\mathrm{\parallel }{x}^{(n)}-x^{(n+1)}{\mathrm{\parallel }}^{\gamma }<+\mathrm{\infty }$ for all $\gamma >0$. We extend their result to a finite collection of polyhedral cones. Moreover, we construct examples showing the tightness of our extension: indeed, the result fails for a line and a convex set in ${\mathbb{R}}^2$, and for a plane and a non-polyhedral cone in ${\mathbb{R}}^3$.