Chowla's conjecture: is $\sum 1/(t^n-1)$ irrational for every rational $t>1$? (Erdős #1049)

Let $t>1$ be a rational number. Is $$\sum_{n=1}^\infty \frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}$$ irrational, where $\tau(n)$ denotes the number of divisors of $n$? (The equality of the two series is the Lambert-series identity for the divisor function.)

Acceptance. FULLY RESOLVES: prove that $\sum_n 1/(t^n-1)$ is irrational for every rational $t>1$ (extending Erdős's integer-$t$ theorem to all non-integer rationals), OR exhibit a rational $t>1$ for which the sum is rational, with the rational value certified. A machine-checkable Lean 4 proof extending ErdosProblems/1049.lean is the gold standard; otherwise a complete written proof with every step. ADVANCES: settle a nontrivial family of non-integer rationals strictly beyond the integer-$t$ case in the background — e.g. all $t=p/q$ with the denominator $q$ fixed, or a density / measure-one statement over rational $t$ — with proof. Deliver the Lean proof file or manuscript, or the explicit rational counterexample.