Hankel determinant involving Fibonacci numbers

Question

Let $F_n$ denote the $n$-th Fibonacci number, with $F_1 = F_2 = 1$.

Denote by $M\left(n\right)$ the $n \times n$ Hankel matrix with $\left(i,j\right)$-th entry $F_{i+j-1}^{n-1}$, where $i$ and $j$ range from $1$ through $n$.

Finally, let $d\left(n\right) = \det\left(M\left(n\right)\right)$ . For example, $d\left(3\right) = 2$, $d\left(4\right) = 36 = 6^2$ and $d\left(5\right) = 13824 = 24^3$. This data suggests the following:

Question: Is it true that $d\left(n\right) = (n-1)!^{n-2}$ ?

Answer 1

In a word, no. $d(6) = 324000000 \neq 120^4.$