I was not able to find a statement of the conjecture by Erdős and Straus in the form stated, but I did find some interesting related work. I recommend that you take a look at the introduction and bibliography (at least) of the paper
Any paper referenced below without a full citation is in the bibliography of Saradha and Tijdeman. This bibliography may provide a useful starting point in trying to locate the conjecture you mention.
The introduction discusses variations on the following conjecture of Erdős, which apparently goes back to 1949 but did not appear in print in Erdős's intended form until a 1965 paper of Livingston.
Conjecture (Erdős). Let $f$ be a number-theoretic function with period $q > 0$ such that $|f(n)| = 1$ for $1\le n < q$ and $f(q) = 0$. Then $ \sum_{n=1}^{\infty} \frac{f(n)}{n} \ne 0 $ whenever the series converges.
In 1973, Baker, Birch, and Wirsing attribute the following problem to Chowla:
Problem. Does there exist a rational-valued function $f(n)$, periodic with prime period $p$, such that $ \sum_{n=1}^{\infty} \frac{f(n)}{n} = 0? $
Baker, Birch, and Wirsing disprove the claim by proving the following theorem.
Theorem. Suppose $f\colon \mathbb{Z}\to \overline{\mathbb{Q}}$ is a nonvanishing function with period $q$. If (i) $f(n) = 0$ whenever $1 < \gcd(r, q) < q$, and (ii) the $q$th cyclotomic polynomial is irreducible over $\mathbb{Q}(f(1),\dots,f(q))$, then $ \sum_{n=1}^{\infty} \frac{f(n)}{n} \ne 0.$
This result and a related result of Okada are used as a basis for Adhikari, Saradha, Shorey, and Tijdeman to prove the following theorem.
Theorem. Suppose $f\colon \mathbb{Z}\to \overline{\mathbb{Q}}$ is periodic with period $q$. If the series $ \sum_{n=1}^{\infty} \frac{f(n)}{n} $ converges to some number $S$, then either $S = 0$ or $S$ is transcendental.
Adhikari et al. comment (I'm paraphrasing here) that applying Baker's theorem tends to lead to dichotomies of the form "$S$ is either rational or transcendental"; beyond that I don't find direct motivation for the claim or prior conjectures of the form you indicate. (But I'm no number theorist.)
Some salt should be ingested with some of the results in Adhikari et al. In particular, I found the following paper which in its introduction claims to provide a counterexample to one of the theorems from that paper: