Sufficiency of the Hautus criterion for exact controllability of difference delay equations

Let $\Lambda_N>0$, $d\in\mathbb{N}$, $q\in[1,+\infty)$, and consider the linear difference delay system with delay parameters and matrices $A_N$ and $B$ described above. Let $H(\mathbb{C})$ denote the corresponding matrix-valued characteristic function, and let $\overline{H(\mathbb{C})}$ be its closure. Hautus sufficiency conjecture. The system is $L^q$ exactly controllable in time $d\Lambda_N$ if and only if

$$\operatorname{rank}[M,B]=d\quad\text{for every }M\in\overline{H(\mathbb{C})},$$

and

$$\operatorname{rank}[A_N,B]=d.$$

The corresponding rank conditions are known to be necessary, while sufficiency would resolve the associated Bézout identity over a Radon measure algebra; this remains an open problem.