Indeed, it is only convex for x^2 \ge a^2/3 (which, although only mattering as a piece of trivia, is itself a non-convex region).
I guess this is another expression for @Erling’s challenge., which, as shown below, does not exclude expressions which are not convex over their entire natural domain. Maybe the challenge should be clarified to include that provision.
https://twitter.com/J_P_Vielma/status/1014513695605608448
Conic modeling challenge by https://twitter.com/e_d_andersen Do you have a nonlinear constraint that can be written in a modeling language, but can’t be represented with linear, socp, sdp, exponential cone and power cones?