Let $E/\mathbb{Q}$ be an elliptic curve and let $N_E(3,X)$ denote the number of cyclic cubic extensions $K/\mathbb{Q}$ of conductor no more than $X$ for which $rank~E(K)> ~rank~ E(\mathbb{Q})$. Then a conjecture of David, Fearnley and Kisilevsky (stemming from considerations in random matrix theory) states that

$ \log N_E(3,X) \sim \frac{1}{2}\log X.$

My question is what the conjecture should be if we remove the condition that $K/\mathbb{Q}$ is a $\textit{cyclic}$ extension.