Infinitely many consecutive Fibonacci divisibility sums

Let $F_i$ denote the $i$th Fibonacci number. Consecutive-pair conjecture. There are infinitely many pairs of positive integers $(n,n+1)$ such that

$$n\mid \sum_{i=1}^n F_i\quad\text{and}\quad n+1\mid \sum_{i=1}^{n+1} F_i.$$

This conjecture asks whether the divisibility property for sums of initial Fibonacci numbers occurs for infinitely many consecutive indices. The supplied text gives no resolution or further evidence, so the conjecture remains open.