A sharper log-convexity inequality for Bell numbers

We prove a stronger version of the log-convexity inequality for the Bell numbers $B_n$. In particular, for $n\ge 5$, we have
\[
B_{n+1}B_{n-1} - (B_n)^2 \ge \sum_{i=1}^{n} F_i (B_{n-i})^2,
\]
where $F_i$ is the $i$-th Fibonacci number with $F_0=F_1=1$.
The simple proof is mostly combinatorial with elementary inequalities.