Arbitrary function on the Faddev-Kulish dressing

On this paper the authors review the Faddev-Kulish dressing in QED which is a solution to the IR divergence problem.

Given one electron momentum $$\mathbf{p}$$, They define the soft factor by

$$F_\ell(\mathbf{k},\mathbf{p})=\dfrac{p\cdot e_\ell(\mathbf{k})}{p\cdot k}\phi(\mathbf{k},\mathbf{p})\tag{1}$$

where $$\ell=1,2$$ labels photon polarizations, $$\mathbf{k}$$ is a photon momentum and $$\phi$$ is any function that smoothly goes to $$\phi\to 1$$ as $$|\mathbf{k}|\to 0$$.

One introduces one IR regulator $$\lambda$$ and an upper infrared cutoff $$E$$ so that any particle with energy less than $$E$$ is deemed soft.

Then they define

$$R_\mathbf{p}=e\sum_\ell\int_{\lambda <|\mathbf{k}|

and define the single electron dressing operator

$$W_\mathbf{p}=\exp R_\mathbf{p}\tag{3}$$

The dressed states are $$\|\mathbf{p}\rangle=W_\mathbf{p} |\mathbf{p}\rangle,\tag{5}$$

and the idea is to work with these dressed states as asymptotic states rather than the free Fock states.

Well, the issue is that from the start we are carrying one arbitrary function $$\phi$$. The function enters the definition of the dressing operator from the start. This is bothering me.

So my question is: do the end results depend on this arbitrary function? If so, what is its origin and how we should understand its arbitrariness?