Stone formula and joint spectral measures for commuting operators

Question

Let $\mathcal{H}$ a Hilbert space and $A$, $B$ two self-adjoint, (possibly) unbounded operators on it with domains $D(A)$, $D(B)$. By the spectral theorem, both operators are uniquely associated with two PVMs $E_A,\,E_B:\mathfrak{B}(\mathbb{R})\rightarrow\mathcal{B}(\mathcal{H})$, the former being the Borel $\sigma$-algebra on $\mathbb{R}$, and with two strongly continuous unitary groups $(e^{-itA})_{t\in\mathbb{R}}$, $(e^{-isB})_{s\in\mathbb{R}}$ satisfying \begin{equation} e^{-itA}=\int_{\mathbb{R}}e^{-itx}dE_A(x),\qquad e^{-isB}=\int_{\mathbb{R}}e^{-isy}dE_B(y). \end{equation} Also suppose that $A$ and $B$ commute, in the sense that their PVMs commute. Under this hypothesis (see for example this question), we can also define joint functions of $A$ and $B$ via a double integral on the two PVMs; in particular, we can define \begin{equation} e^{-it(A-B)}=\iint_{\mathbb{R}^2}e^{-it(x-y)}dE_A(x)\,dE_B(y). \end{equation} Now, suppose that the operator $A-B$, defined on $D(A-B)=D(A)\cap D(B)$, is (at least essentially) self-adjoint itself. Then there exists a PVM $E_{A-B}:\mathfrak{B}(\mathbb{R})\rightarrow\mathcal{B}(\mathcal{H})$ such that \begin{equation} e^{-it(A-B)}=\int_{\mathbb{R}}e^{-itz}dE_{A-B}(z). \end{equation} What is the link between the PVMs $E_{A-B}$, $E_A$ and $E_B$? A naive comparison between the two formulas above seems to suggest that $E_{A-B}$ is a sort of marginal of the joint PVM of $A$ and $B$, i.e., formally, \begin{equation} dE_{A-B}(z)=\int_{\mathbb{R}}dE_A\left(\frac{z+z'}{2}\right)dE_B\left(\frac{z-z'}{2}\right), \end{equation} but I do not know whether this identity actually holds, nor how to prove it.