Maximum-gap sampling conjecture for shift-invariant spline spaces

Let $Q_m$ denote the spline generator and let $V(Q_m)$ be the shift-invariant spline space it generates. A sequence $\\{x_i:i\in\mathbb{Z}\\}$ is separated if its distinct points have a positive minimum separation. Maximum-gap sampling conjecture. If $\\{x_i:i\in\mathbb{Z}\\}$ is a separated set with

$$\sup\limits_{i}(x_{i+\nu}-x_i)<\nu,$$

then $\\{x_i:i\in\mathbb{Z}\\}$ is a stable set of sampling for $V(Q_m)$. This extends the known case $\nu=1$ and is motivated by the sharp maximum-gap results for entire functions of exponential type $\pi$ and the shift-invariant space generated by the Meyer scaling function; the corresponding assertion for general $\nu\geq 1$ remains open.