The common-ground conjecture on strong standard canonical form for regular DAEs

Let $E,F:cal I\rightarrow\mathbb{R}^{m\times m}$ be sufficiently smooth matrix functions, and consider the pair $\{E,F\}$ in the differential-algebraic equation $Ex'+Fx=q$. Assume that the pair is regular, with constant rank $r=\operatorname{rank}E(t)<m$, index $\mu\geq 2$, and canonical characteristics $\theta_0\geq\cdots\geq\theta_{\mu-2}>\theta_{\mu-1}=0$. A pair is in strong standard canonical form when it is equivalent to

$$\left\{\begin{bmatrix}I_d&0\\0&N\end{bmatrix},\begin{bmatrix}\Omega&0\\0&I_{m-d}\end{bmatrix}\right\},$$

where $N$ is a constant nilpotent matrix and $d=r-\sum_{i=0}^{\mu-2}\theta_i$.

The common-ground conjecture. The pair $\{E,F\}$ is transformable into strong standard canonical form, and the Jordan normal form of its constant nilpotent matrix $N$ consists exactly of $m-r-\theta_0$ Jordan blocks of order $1$, $\theta_0-\theta_1$ blocks of order $2$, $\theta_1-\theta_2$ blocks of order $3$, continuing with $\theta_{\mu-3}-\theta_{\mu-2}$ blocks of order $\mu-1$, and $\theta_{\mu-2}$ blocks of order $\mu$.

The claim supplies a canonical matrix description of regular linear time-varying DAEs and identifies the invariant characteristics with the sizes and multiplicities of the nilpotent Jordan blocks. The supplied paper states that it proves this conjecture, so its status is recorded as solved.