Irrationality of $\sum 1/F_{n_k}$ for lacunary Fibonacci subsequences with ratio $c\in(1,2)$ (Erdős #267)

Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci numbers, and let $n_1<n_2<\cdots$ be an infinite increasing sequence of indices satisfying $n_{k+1}/n_k\ge c$ for a fixed constant $c>1$. Must $$\sum_k \frac{1}{F_{n_k}}$$ be irrational? (The case $c\ge 2$ is known; the open range is $1<c<2$.)

Acceptance. FULLY RESOLVES: prove that for every $c\in(1,2)$ and every index sequence with $n_{k+1}/n_k\ge c$ the sum $\sum_k 1/F_{n_k}$ is irrational (closing the gap left by Badea's $c\ge 2$ theorem), OR exhibit a specific $c\in(1,2)$ and an explicit index sequence with $n_{k+1}/n_k\ge c$ for which the sum is rational, with the rational value certified. A machine-checkable Lean 4 proof extending ErdosProblems/267.lean is the gold standard; otherwise a complete written proof with every step. ADVANCES: lower the ratio threshold below 2 — prove irrationality for all $c\ge c_0$ with an explicit $c_0<2$, strictly improving on the $c\ge 2$ bound stated in the background, with proof; or settle the weaker sufficient condition $n_k/k\to\infty$. Deliver the Lean proof file or manuscript, or the explicit rational counterexample.