There's a famous trick attributed to Minty (sometimes also to Browder and Lions) which allows to recover at least a.e. convergence when you're able to pass to the limit through an increasing non-linearity (typically $x\mapsto x^p$). As I guess you read french (right ?) you can check " Exercice 2 " [ in that document][1]. 

I don't know if's applicable in your example but it is worth giving a shot (and it's friday night, so I let you check !). I would believe avoiding the regions in which $(\rho_n)_n$ gets too closer to $0$ (which are actually OK as far oscillations are concerned), you should be able to recover an increasing setting for your non-linearity.

  [1]: https://www.ljll.math.upmc.fr/~moussa/pdf/enseignements/td_af_1_bonus.pdf