Middle-degree BRST cohomology splitting conjecture for the bosonic ambitwistor string

Let $p$ be a momentum, let $H$ be its stabiliser, and let $\mathsf{H}^n(p)$ and $\mathsf{H}_{\mathrm{rel}}^n(p)$ denote the absolute and relative BRST cohomology groups at ghost number $n$. Write $\mathsf{H}_{\mathrm{rel}}^2(p)^*$ for the dual $H$-module. Middle-degree BRST splitting conjecture. The short exact sequence relating $\mathsf{H}_{\mathrm{rel}}^2(p)$, $\mathsf{H}^3(p)$, and $\mathsf{H}_{\mathrm{rel}}^3(p)$ should split as an $H$-module, so that

$$\mathsf{H}^3(p) \cong \mathsf{H}_{\mathrm{rel}}^2(p) \oplus \mathsf{H}_{\mathrm{rel}}^2(p)^* \cong \mathsf{H}^2(p) \oplus \mathsf{H}^2(p)^*.$$

This is proposed as the non-unitary analogue of the corresponding splitting in the unitary $K$-module setting. The authors state that they have no evidence beyond this naive generalisation.