Super-diffusive drift conjecture for bounded-degree Cayley graphs of symmetric groups

Let $d_{T_n}$ be the word metric on the Cayley graph $\Gamma(\operatorname{Sym}_n,T_n)$. For the simple random walk $(W_t)_{t=1}^{\infty}$ on this graph started at $W_0=1$, write $\mathbb{E}$ for expectation. Super-diffusive drift conjecture. For every $r\in\mathbb{N}$ and $M>0$, there exists $n\in\mathbb{N}$ such that, for every generating set $T_n$ of $\operatorname{Sym}_n$ with $|T_n|\leq r$,

$$\sup_{t\in\mathbb{N}}\frac{\mathbb{E}\,d_{T_n}(1,W_t)^2}{t}>M.$$

If true, this would imply a negative answer to the question of whether symmetric groups admit uniformly Hilbert-embeddable Cayley graphs with uniformly bounded degree. The source presents the assertion as a conjecture and gives no resolution status.