Canonical Jordan block structure conjecture for regular differential-algebraic equations

Let $E,F:\mathcal I\rightarrow\mathbb{R}^{m\times m}$ be sufficiently smooth, and let $\{E,F\}$ be a regular pair with index $\mu\geq 2$. Write $r$ for the dimension of the differential part and $\theta_0\geq\cdots\geq\theta_{\mu-2}>\theta_{\mu-1}=0$ for its characteristic values. In strong standard canonical form, let $N$ denote the nilpotent matrix associated with the algebraic part. Canonical Jordan block structure conjecture. The pair $\{E,F\}$ is transformable into strong standard canonical form, and the Jordan normal form of $N$ consists exactly of

$$m-r-\theta_0$$

Jordan blocks of order $1$,

$$\theta_0-\theta_1$$

Jordan blocks of order $2$, continuing with

$$\theta_{\mu-3}-\theta_{\mu-2}$$

Jordan blocks of order $\mu-1$, and

$$\theta_{\mu-2}$$

Jordan blocks of order $\mu$. This conjecture would provide a canonical justification for the characteristic values of regular differential-algebraic equations beyond the constant-matrix case; its status is not determined by the supplied text.