I have one probably very silly confusion about a footnote in the paper "2D Kac-Moody symmetry of 4D Yang-Mills theory ". In section (4) the authors consider ${\cal O}_k(E_k,z_k,\bar{z}_k)$ an operator which creates or annihilates a colored hard particle with energy $E_k\neq 0$ crossing the $S^2$ on ${\mathscr{I}}$ at the point $z_k$. In a footnote they say that for scalar particles we would have: $${\cal O}_k(E_k,z_k,\bar{z}_k)=-\frac{4\pi}{E_k}\int_{-\infty}^\infty du e^{iE_k u}\partial_u \lim_{r\to\infty}[r\phi_k(u,r,z_k,\bar{z}_k)]\tag{1}.$$
Now the way I understood this is that ${\cal O}_k(E_k,z_k,\bar{z}_k)$ is just one creation/annihilation operator written in terms of the field data at $\mathscr{I}$. I have tried to take one large $r$ limit of a scalar field and obtain (1).
In that case I have considered the simplest example possible: one massless scalar field $\phi(x)$. Decomposing into creation and annihilation operators $$\phi(x)=\int\dfrac{d^3 p}{(2\pi)^32\omega} (a(p)e^{ipx}+a^\dagger(p)e^{-ipx}),\tag{2}$$
I considered the $r\to \infty$ limit with $(u,z,\bar{z})$ fixed employing the plane wave decomposition into spherical Bessel functions plus the asymptotic behavior of such functions. As a result I have obtained $$\phi(u,r,z,\bar{z})=-\dfrac{i}{8\pi^2 r}\int_0^\infty [a(\omega\hat{x}(z,\bar{z}))e^{-i\omega u}-a^\dagger(\omega\hat{x}(z,\bar{z}))e^{i\omega u}] d\omega+O\left(\frac{1}{r^2}\right)\tag{3}.$$
Now using (1) the result is exactly $a(\omega\hat{x}(z,\bar{z}))$. So it seems to confirm that ${\cal O}$ is really just the familiar creation/annihilation operators, just written in terms of ${\mathscr{I}}$ data.
But if that is the whole point (write the creation/annihilation operators in terms of ${\mathscr{I}}$ data) then why instead of dividing by the energy and taking $\partial_u$ we don't just take $${\cal O}(\omega,z,\bar{z})=4\pi i \int_{-\infty}^\infty e^{i\omega u}\lim_{r\to \infty}(r\phi(u,r,z,\bar{z}))du\tag{4}.$$
I mean (4) does the same job and it seems more natural. So is there any reason to use (1) instead? Why use (1) instead of (4)?