The first cohomology conjecture for differential operators on the supercircle

Let $\mathrm{osp}(1|2)$ act on the space ${\cal D}_{\lambda,\mu}$ of differential operators from tensor densities of weight $\lambda$ to tensor densities of weight $\mu$, and let $H^1(\mathrm{osp}(1|2);{\cal D}_{\lambda,\mu})$ denote its first cohomology space. For each odd integer $k$, let $\gamma_k$ be the cocycle with values in ${\cal D}_{\frac{1-k}{4},\frac{1+k}{4}}$. First cohomology conjecture. One has

$$H^1(\mathrm{osp}(1|2);{\cal D}_{\lambda,\mu})=\mathbb{C}^{0|1}$$

if and only if $(\lambda,\mu)=(\frac{1-k}{4},\frac{1+k}{4})$ for an odd integer $k$, with the cohomology spanned by the cocycle $\gamma_k$. Otherwise, this cohomology space is trivial. The conjecture proposes a complete description of the first cohomology of $\mathrm{osp}(1|2)$ with coefficients in differential operators on the supercircle; the supplied text does not establish the claim, so its resolution remains open.