Critical points of the two-dimensional Ambrosio-Tortorelli functional with convergence of the phase-field energy

We consider a family $\{(u_\varepsilon, v_\varepsilon)\}_{\varepsilon>0}$ of critical points of the Ambrosio-Tortorelli functional. Assuming a uniform energy bound, the sequence $\{(u_\varepsilon, v_\varepsilon)\}_{\varepsilon>0}$ converges in $L^2(Ω)$ to a limit $(u, 1)$ as $\varepsilon \to 0$, where $u$ is in $SBV^2(Ω)$. It was previously shown that if the full Ambrosio-Tortorelli energy associated to $(u_\varepsilon,v_\varepsilon)$ converges to the Mumford-Shah energy of $u$, then the first inner variation converges as well. In particular, $u$ is a critical point of the Mumford-Shah functional in the sense of inner variations. In this work, focusing on the two-dimensional setting, we extend this result under the sole convergence of the phase-field energy to the length energy term in the Mumford-Shah functional.