This guess must fail for large $n$ because $d(S_n)$ grows much faster than $n(n-1)/2$. Indeed $d(S_n) > n(n-1)/2$ for all $n \geq 16$ (and possibly also a few smaller $n$).

For any group $G$, a lower bound on $D(G)$ is the maximal order $|a|$ of any element $a \in G,$ because the sequence $a, a, a, \ldots$ of length $|a|-1$ is one-free. For $G = S_n$, this lower bound (Landau's function $g(n),$ OEIS sequence A000793) grows faster than $n(n-1)/2$ and indeed faster than any polynomial in $n$. Already for $n=16$ the product of disjoint cycles of lengths $4,5,7$ has order $140$ while $16(16-1)/2 = 120$. The OEIS goes just far enough to see that $D(47) \geq 120120 > 47^3$.

This guess must fail for large $n$ because $d(S_n)$ grows much faster than $n(n-1)/2$. Indeed $d(S_n) > n(n-1)/2$ for all $n \geq 16$ (and possibly also a few smaller $n$).

For any group $G$, a lower bound on $D(G)$ is the maximal order $|a|$ of any element $a \in G,$ because the sequence $a, a, a, \ldots$ of length $|a|-1$ is one-free. For $G = S_n$, this lower bound (Landau's function $g(n),$ OEIS sequence A000793) grows faster than $n(n-1)/2$ and indeed faster than any polynomial in $n$. Already for $n=16$ the product of disjoint cycles of lengths $4,5,7$ has order $140$ while $16(16-1)/2 = 120$. The OEIS goes just far enough to see that $D(S_{47}) \geq 120120 > 47^3$.