Restricted-isometry conjecture for truncated sparse approximation properties

Let $A$ be a measurement matrix, let $k$ be a positive integer, and let $t\in(0,4/3)$. Write $\delta_{tk}$ for the restricted 2-isometry constant of order $tk$. The $(l_2,l_1)$-$l_2$ sparse approximation property and the $(l_2,l_1)$-Dantzig selector sparse approximation property are the properties of order $k$ defined for the corresponding truncated minimization models.

Restricted-isometry conjecture. If $A$ has restricted 2-isometry property of order $tk$ with

$$\delta_{tk}<(4-t)/t,$$

then $A$ satisfies $(l_2,l_1)$-$l_2$ sparse approximation property of order $k$ with certain constants $0<\gamma<1$ and $0<D_3<\infty$, and $(l_2,l_1)$-Dantzig selector sparse approximation property of order $k$ with certain constants $0<\gamma<1$ and $0<D_4<\infty$.

This conjecture proposes a sufficient restricted-isometry condition for stable sparse approximation under both truncated $l_2$ minimization and truncated Dantzig-selector minimization in the previously unresolved range $0<t<4/3$.