The strong weight complex functor conjecture

Let $(\underline{C},w)$ be a triangulated category with a weight structure $w$, let $\underline{Hw}=\underline{C}^{w=0}$ be its heart, and let $t:\underline{C}\to K_{\mathfrak{w}}(\underline{Hw})$ be the weak weight complex functor. Strong weight complex functor conjecture. The functor $t$ can be lifted to an exact functor

$$t^{st}:\underline{C}\to K(\underline{Hw}).$$

Such a lift would strengthen the weak weight complex construction by placing it in the ordinary homotopy category of complexes; the paper notes that the lift exists in important differential graded settings, but leaves the general case conjectural.