Existence of suitable decay for cube-based CIS interpolation in higher dimensions

Let $n atural$ be a positive integer. A complete interpolating sequence (CIS) for cubes is a sampling sequence that also permits interpolation for the relevant band-limited function space. For $\alpha>0$, define

$$g_\alpha(x)=\int_{\mathbb{R}^n}e^{-\alpha|\xi|^{\omega}}e^{i\langle \xi,x\rangle}\,d\xi.$$

The cube-CIS recovery conjecture. There exists $\omega(n)>0$ such that the interpolation operator $I_\alpha$ associated to $g_\alpha$ admits the use of CISs for cubes in $\mathbb{R}^n$ to recover functions as in Theorems 2 and 3. The conjecture proposes a kernel with slower decay that would extend the Poisson interpolation procedure from two dimensions to cubes in higher dimensions. The supplied text gives no resolution, so the conjecture remains open.