Why $\int_{\mathcal I_+^+} \varepsilon*F=0$ for any $\varepsilon$ when there are no massive charges? My problem is really simple. I was reading the Strominger lectures where he defines the future charges $Q_\varepsilon^+$, and he does something that I don't understand. He says on the equation $(2.5.4)$ that without massive charges the following affirmation is true:
$$
I=\int_{\mathcal I_+^+}
\varepsilon*F=0
$$
Where $F$ is the Electromagnetic Tensor, $\,\mathcal I^+$ is the Future Null Infinity, $\,\mathcal I^+_-$ is its past and $\,\mathcal I^+_+$ is its future. And the only condition for $\varepsilon$ is that it obeys $\varepsilon\big|_{\mathcal I^+_-}=\varepsilon\big|_{\mathcal I^-_+}$
The reason for this affirmation according to the book is that the electric field will vanish at $\mathcal I^+_+$. I don't know why these affirmations are true.

My attempts:
Attempt 1: If there is no charges, according to Maxwell equations:
$$
d*F=0\implies *F=d\omega
$$
Where $\omega$ is a $1$-form. Then the integral is:
$$
I=\int_{\mathcal I_+^+}
\varepsilon \,d\omega
$$
Using $d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^p\alpha\wedge d\beta$ where $\alpha$ is a $p$-form:
$$
I=\int_{\mathcal I_+^+}
\Big[
d(\varepsilon\omega)
+\omega\wedge d\varepsilon
\Big]
$$
And I suppose I could do something with the generalized Stokes theorem or something like that. But I really have no clue.
Attempt 2: I kind of have an intuition, maybe it is wrong, $\mathcal I^+_+$ is like a $2$-manifold (on spacetime) of size going to zero, so all the integrals of the form $\int_{I^+_+}$ will be zero.
Any help or advice would be awesome.

References:
Lectures on the Infrared Structure of Gravity and Gauge Theory - Andrew Strominger
Lectures on Youtube
 A: Not 100% sure but here is my guess. Somehow the fact that the particle is massless must come into play. From equation 2.3.2
$$
F_{rt}\rvert_{\mathcal{I}^+} = \frac{e^2}{4 \pi r^2} \sum_{k=1}^n \frac{Q_k}{\gamma_k^2 (1 - \hat x \cdot \vec{\beta}_k )^2}.
$$
When computing $\star F$ over an $S^2$ integral at $\mathcal{I}^+_+$, only the $\phi$ and $\theta$ components of $\star F$ come into play. This is the same as the $r$ and $t$ components of $F$. Note that $F_{rt}$ above goes as $1/r^2$ while our spherical integral will go as $r^2$, so a priori we will have a $\mathcal{O}(1)$ left over. So something about this expression must go to zero if the particles are massless. I think it is $\gamma_k$ of the particles. If $|\beta_k| = 1$ then $\gamma_k \to \infty$ and $F_{rt} \to 0$.  (However, I think you also have to check that $(1 - \hat x \cdot \vec{\beta}_k)^2$ won't somehow mess this up. Also, would this mean that the expression is zero on the whole $\mathcal{I}^+$, save for maybe a discrete set of $\hat x$?)
Also, if you continually struggle to make sense of the book, I think some of the original papers go into more detail and may be a bit clearer on some technical points. A quick arxiv search reveals the following promising looking results.
https://arxiv.org/abs/1506.02906
https://arxiv.org/abs/1412.2763
https://arxiv.org/abs/1407.3814
https://arxiv.org/abs/1407.3789
