The second cohomology structure conjecture for differential operators on differential forms

Let $\mathrm{Vect}(\mathbb R^n)$ be the Lie algebra of vector fields on $\mathbb R^n$, let $\Omega(\mathbb R^n)$ denote the space of differential forms, and let ${\mathcal D}^{k,\ell}$ be the space of differential operators from $k$-forms to $\ell$-forms. The second cohomology is $H^2(\mathrm{Vect}(\mathbb R^n);{\mathcal D}^{k,\ell})$. Second cohomology structure conjecture. The space of second cohomology has the structure

$$H^2(\mathrm{Vect}(\mathbb R^n);{\mathcal D}^{k,\ell})= \left\{ \begin{array}{ll} \mathbb R,& \ell=k+1,\\ \mathbb R^2,& \ell=k+2,\\ \mathbb R,& \ell=k+3,\\ 0,& \text{otherwise}, \end{array} \right.$$

and is spanned by the $2$-cocycles $\gamma^k_1$, $\gamma^k_2$, $\widetilde{\gamma}^k_2$ and $\gamma^k_3$ given by the formulas referred to in the source. This conjecture seeks a complete description of the second cohomology ring's degree-two part; the surrounding text notes that only the first cohomology was previously completely known and that the displayed cocycles provide a lower bound for the second cohomology.