Conjecture: Binomial sum congruence on Fibonacci and prime numbers

My experimental data suggests that the following binomial sum congruence involving primes and Fibonacci numbers hold. I have experimentally verified them for all primes $\le 1.5 \times 10^6$. Can they be proved or disproved?

Conjecture 1. Let $p$ be a prime, $ p\equiv 1 \text{ or } 19 \pmod{30}. $ Then $$ \sum_{k=0}^{p-1} \binom{2k}{k}^{3} \frac{F_{8k}}{4096^k} \equiv 0 \pmod{p^3}, $$ where $F_n$ denotes the $n$th Fibonacci number.

Conjecture 2. Let $p>5$ be a prime satisfying $ p\equiv 1,9,11,\text{ or }19 \pmod{40}. $ Then $$ \sum_{k=0}^{p-1} \binom{2k}{k}^{3} \frac{(-1)^kF_{12k}}{64^k} \equiv 0 \pmod{p^3}, $$ where $F_n$ denotes the $n$th Fibonacci number.