Optimal average Sobolev-norm growth bound for generic Gaussian Euler data

Let $c3>0$ and let $(a_n)_{n\in\mathbb{Z}^2}\in h^{\sigma}$ be a sequence that is not exclusively supported on a line through the origin or on a circle centered at the origin. Let $a9_0^\omega$ be initial data sampled from the Gaussian construction associated with $(a_n)$, and let $a9^\omega(t)$ be the corresponding Euler-flow solution. For $s\in(0,\sigma]$, write $t_0=t_0((a_n),s)$.

Optimal growth conjecture. There exists $C(s)>0$ such that, for every $t\in[0,t_0)$ and every $s\in(0,\sigma]$,

$$\mathbb{E}\left\|\Omega^\omega(t)\right\|_{H^s}\geq\mathbb{E}\left\|\Omega_0^\omega\right\|_{H^s}+Ct^2.$$

This is presented as an optimality question for the available Yudovich growth bound. The stated lower bound is open in the indicated generality, so it is not known whether the quadratic average-growth estimate holds for all such non-exceptional Gaussian data.