Zero-momentum BRST cohomology conjecture for the bosonic ambitwistor string

Let $\mathsf{H}^n(0)$ denote the absolute BRST cohomology at ghost number $n$ and zero momentum, regarded as a representation of $\operatorname{SO}(25,1)$, and let $\mathbb{V}$ be the vector representation. Zero-momentum BRST cohomology conjecture.

$$\mathsf{H}^n(0) \cong \begin{cases} \mathbb{C} & n=0,6\\ \mathbb{C} \oplus 2 \mathbb{V} & n=1,5\\ 2 \mathbb{C} \oplus 2 \mathbb{V} \oplus \bigodot^{\!2}_0 \mathbb{V} \oplus \bigwedge^{\!2} \mathbb{V} & n=2,4\\ 4 \mathbb{C} \oplus 2 \bigodot^{\!2}_0 \mathbb{V} \oplus 2 \bigwedge^{\!2} \mathbb{V} & n=3.\end{cases}$$

The result follows from the computed relative cohomology together with the conjectural Poincaré duality of the absolute BRST cohomology at zero momentum; the degrees not directly determined by the calculation therefore remain conditional.