Explicit splitting conjecture for truncated de Rham cohomology of the Drinfeld space

Let $G$ be the group acting on the Drinfeld space, let $D(G)$ be its distribution algebra, and let $\Omega^\bullet$ be the de Rham complex. For $\ell\in\{2,\dots,n-2\}$, write $\tau_{\leq\ell}\Omega^\bullet$ for the truncation and $H^\ell(\Omega^\bullet)[-\ell]$ for the shifted $\ell$-th cohomology complex. A section is sought in the derived category of finite length coadmissible $D(G)$-modules with Orlik–Strauch irreducible constituents.

Explicit splitting conjecture. For $\ell\in\{2,\dots,n-2\}$, the morphism of complexes

$$\tau_{\leq\ell}\Omega^\bullet\twoheadrightarrow H^\ell(\Omega^\bullet)[-\ell]$$

admits a section in the derived category of finite length coadmissible $D(G)$-modules with Orlik–Strauch irreducible constituents.

Such sections would extend the explicit splittings already known in degrees $0$ and $1$ and would yield a full explicit splitting of the de Rham complex in the corresponding range. The paper gives a construction in the case $n=4$, $\ell=2$, but states the general existence claim as a conjecture.