I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations).

Let A,B be two hearts of bounded t-structures in a triangulated category D. Let E be an object. Then there is an $E_2$ spectral sequence $H^p_A( H^q_B(E)) \Rightarrow H_A^{p+q}(E)$.

Where $H^i_\bullet$ is the cohomology with respect to the t-structure $\bullet$. The point of the spectral sequence is that you can filter E in two ways, according to A or to B and then A.