A shift-extension conjecture for Fibonacci divisibility sums

Let $F_i$ denote the $i$th Fibonacci number, and for a positive integer $m$ write $\sum_{i=1}^m F_i$ for the sum of the first $m$ Fibonacci numbers. Shift-extension conjecture. For each positive integer $t$, there is a positive integer $n$ such that

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

This conjecture concerns the distribution of positive integers that divide their corresponding sums of initial Fibonacci numbers. The supplied text gives no resolution or further evidence, so the conjecture remains open.