A nearly tight bound on the probability $G(n,p)$ is even-degenerate

A graph is even-degenerate if we can order its vertices as $v_1,\dots,v_n$ so that the number of edges from $v_i$ to $v_1,\dots,v_{i-1}$ is even for all $i>2$. We show that the probability $G(n,p)$ is even-degenerate is $e^{-Ω(n/\log\log n)}$, which is only a factor of $\log\log n$ away, in the exponent, from the trivial lower bound $2^{-O(n)}$.