Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles functor $\Phi_{f_i}$ and the Beilinson's vanishing cycles functor $\Phi'_{f_i}$. For a perverse sheaf $F$ (of $k$-vector spaces) on $X$, the objects $\Phi_{f_i}(F)$ and $\Phi'_{f_i}(F)$ are isomorphic, but it's seemingly not known whether the isomorphism can be made canonical.
Let $Z_i$ denote the vanishing loci of $f_i$ with the embeddings $w_i: Z_i\to X$ and complementary embeddings $u_i: X\setminus Z_i\to X$. I will also need the open embeddings $j_i: Z_i\setminus Z_1\cap Z_2 \to Z_i$. I have already checked that there are canonical isomoprhisms $\Phi_{f_1\circ i_2}j_2^*\Psi_{f_2}\cong\Phi_{f_2\circ i_1}\Psi_{f_1}u_2^*$ and similarly for the switch of two $\Psi$ functors or two $\Phi$ functors (with some restriction to an open subset put in when needed). The question is whether the same thing is true for the interchange of $\Psi$ and $\Phi'$, which I failed to prove.
The motivation is this: I am interested in the category of perverse sheaves on $X$ constructible with the respect to the stratification which corresponds to the filtration $Z_1\cap Z_2\subset Z_1\cup Z_2\subset X$. In my case I understand the perverse sheaves on each $Z_i$ and on each complement of $Z_i$, but don't understand the vanishing cycle functors $\Psi_{f_i}$. If I could switch the nearby cycles and the Beilinson's vanishing cycles, I could maybe glue the whole category from perverse sheaves on $Z_1$ and its complement $U_1$ (using $\Phi'$ and $\Psi$), write each of these categories as gluing data themselves, and reduce to computing various functors on the parts that I know.
I think this should be known. For instance, when people write down the description of perverse sheaves $\mathbb{C^2}$ with the normal crossings stratification, they seem to have no issue saying that it's equivalent to the category of quadruples of vector spaces computed as the four possible combinations of $\Phi$ and $\Psi$. I am fine with this in this case, but the case of 4-stratum stratifications of a general $X$ still eludes me.
