We study the free energy of mixed $p$-spin spin glass models enriched with an additional magnetic field given by the canonical Gaussian field associated with a Ruelle probability cascade. We prove the conjecture in  that this free energy converges to the Hopf-Lax solution of a certain Hamilton-Jacobi equation. Using this result, we give a new representation of the free energy of mixed $p$-spin models with soft spins.
"Extending the Parisi formula along a Hamilton-Jacobi equation." Electron. J. Probab. 25 1 - 17, 2020. https://doi.org/10.1214/20-EJP432