The extended thin elements conjecture for Kan completions

Let $K$ be a precubical set, let $F(K)$ be the free non-contracting $\omega$-category generated by $K$, and let $\widehat{F(K)}$ denote its Kan completion. Let $C^{gl}$ and $C^{\pm}$ be the associated unnormalized chain complexes, and let $C R^{gl}$ and $C R^{\pm}$ be their quotients by thin elements. The extended thin elements conjecture. The chain complex morphisms

$$C^{gl}(\widehat{F(K)})\rightarrow C R^{gl}(\widehat{F(K)})$$

and

$$C^{\pm}(\widehat{F(K)})\rightarrow C R^{\pm}(\widehat{F(K)})$$

induce isomorphisms in homology. This extends the thin-elements assertion from free precubical $\omega$-categories to their Kan completions; the source gives no resolution status.