Higher truncation functors and factorization conjecture for weight structures

Let $(\underline{C},w)$ be a triangulated category with a weight structure, let $N\geq 0$, and write $\underline{C}^{[0,N]}$ for the corresponding weight range. The strong weight complex functor is denoted $t_0$. Higher truncation conjecture. For every such $(\underline{C},w)$ there exist exact higher truncation functors

$$t_N:\underline{C}\to \underline{C}_N,$$

with $t_0$ the strong weight complex functor, such that for $X,Y\in\underline{C}^{w=0}$,

$$\underline{C}_N(t_N(X),t_N(Y)[-i])=\underline{C}(X,Y)\quad(0\leq i\leq N),$$

and this group is zero otherwise; moreover, there are full embeddings $i_N:\underline{C}^{[0,N]}\to\underline{C}_N$ that send the specified distinguished triangles to distinguished triangles. Finally, if $I:\underline{C}\to D(A)$ is exact and $I(\underline{C}^{w=0})\subset D_{[0,N]}(A)$, then $I$ factors through $t_N$. These conjectures seek higher analogues of the strong weight complex functor and a universal factorization property for exact functors with bounded cohomological amplitude; the paper does not prove them in general.