At the end of my article Hyperelliptic action integral, Annales de l'institut Fourier 49(1), p.303â€“331, I state the following conjecture:

Let D be the open unit disk in the complex plane and $U_1,U_2,\,\dots\,,U_n$ be a cover of the punctured disk D*= D\{0} by open connected sets. Suppose on each open set $U_j$ there is an injective holomorphic function $f_j$ such that $df_j=df_k$ on every intersection $U_j\cap U_k$. Then those differentials glue together to a meromorphic 1-form on D.

It is evident that the 1-form is holomorphic on D*. In the case that its residue at the origin vanishes, the conjecture follows because the singularity can be at most a pole, as can easily be derived from Picard's big theorem (cited below). But I am unable to prove the conjecture for the case of a non-zero residue.
I would be grateful for any proof or counter-example — well, trule spoken, a counter-example would be less welcome since I hope that the conjecture is true. I am guided by my simple geometric intuition on Riemann surfaces...

In 1880 Charles Emile Picard (1856-1941) proved the following theorem.

Picard's Big Theorem
A holomorphic function takes in any neighborhood of an essential singularity every complex value, with at most one exception, infinitely often.

A typical example for Picard's theorem

The function defined by
$\:f(z)=e^{1/z}=\sum_{k=0}^{\infty}\:\frac1{k!z^k}\;$

is holomorphic on $\mathbb{C}\backslash0$ and has an essential singularity in $0$. Is there a value that is does not take (Picard says "at most one exception")? Yes, since $f(z)\neq0$ for all $z\in\mathbb{C}\backslash0$, there is an exceptional value and it must be zero; according to Picard's theorem there are, for any complex value $w\neq0$ and any $\epsilon>0$, infinitely many complex numbers $z$ such that $0<|z|<\epsilon$ and $f(z)=w$.

Direct compuation of this example

In fact, for the example above the theorem of Picard is not really necessary, because we can also see what happens by direct computation: let $w$ be a non-zero complex value and let $\epsilon>0.$ There existe two real numbers $r>0$ and $\varphi$ such that
$w=re^{i\varphi}.$

For $n \in \mathbb{N}$ let $u_n=\ln r+i(\varphi+2\pi n)$ and $z_n=1/{u_n}.$ Thus we have $\lim_{n\to\infty}z_n=0.$
Hence
$f(z_n)=e^{u_n}=e^{\ln r+i(\varphi+2\pi n)}=re^{i \varphi}=w.$

By taking $n$ big enough we see that $w$ has infinitely many pre-images in the punctured disk $0<\,|z|\,<\epsilon\,.$

A less simple example

Denote by P the set of all prime numbers and consider the following function

$g(z)=\sum_{p \in P}^{}\frac{1}{p!z^p}\;.$

Since there is an essential singularity we can apply Picard's Theorem.
But a direct computation seems impossible...