$\mathrm{Diff}(D^4 \text{ rel } \partial)$ is path-connected

Let $\mathrm{Diff}(D^4)$ denote the group of smooth diffeomorphisms of the closed $4$-ball that restrict to the identity on the boundary sphere, with the $C^\infty$ topology. The open question singled out in the source is whether this group is connected. The goal asserts that it is:

Since $\mathrm{Diff}(S^4)\simeq O(5)\times\mathrm{Diff}(D^4)$, this is equivalent to $\pi_0\,\mathrm{Diff}(S^4)\cong\mathbb Z/2$, i.e. every orientation-preserving diffeomorphism of $S^4$ is smoothly isotopic to the identity. The status is open: a proof settles the question positively, a disproof (exhibiting a diffeomorphism not isotopic to the identity rel boundary) settles it negatively.

Formalization Note DiffDisk 4 is the group of diffeomorphisms of $\mathbb R^4$ equal to the identity at every point of norm $\ge 1$, with the topology of uniform convergence of all derivatives; this is the standard model of $\mathrm{Diff}(D^4\text{ rel }\partial)$. For a topological space, path-connected means nonempty with any two points joined by a path, i.e. $\pi_0$ is a single point.