If $\mathcal{H}$ is the heart of a bounded t-structure in a triangulated category $\mathcal{T}$, then for every object $E$ in $\mathcal{T}$ there exists a finite sequence of integers $k_1>k_2>\dots >k_n$ and a collection of triangles in $\mathcal{T}$ $$ E_{i-1}\stackrel{\phi_i}{\to}E_i\to A_i \to E_{i-1}[1] $$ for $i=0,\dots,n$, with $E_0=0$, $E_n=E$ and $A_i\in \mathcal{H}[k_i]$ for any $i=1,\dots,n$. This can be seen as a Postnikov tower for the initial morphism $0 \to E$, so by analogy with what happens in algebraic topology one is led to wonder whether more generally one has Postnikov towers for arbitrary morphisms in a bounded t-structure, i.e., the same as above, but this time with $E_0=X$, $E_n=Y$ and $\phi_n\circ\phi_{n-1}\circ\cdots\circ \phi_1$ equal to some given morphism $f:X\to Y$.

I have so far been unable to rigorously prove (or disprove) such a statement nor to locate it in the literature. As usual any suggestion in either direction will be appreciated.