We investigate the properties of local minima of a recently introduced spin glass model of soft spins subjected to an anharmonic quartic local potential which serves as a model of low temperature molecular or soft glasses. We track the long time gradient descent dynamics in the glassy phase through dynamical mean field theory and show that spins are separated in two groups depending on their local stiffness. For spins having local stiffness that is right above its smallest possible value, the local fields distribution displays a depletion around the origin while those having a stiffness right below its largest possible value have a regular local fields distribution. We rationalize these findings through the replica method and show that the finite temperature phase transition to the glass phase is of continuous (full) replica-symmetry-breaking (RSB) type at low temperatures, down to zero temperature. Furthermore, marginal stability of the zero temperature fullRSB solution implies a linear pseudogap in the density of cavity fields for the spins with a local effective stiffness that is below a certain threshold. This generates a hole around the origin in the corresponding local field distribution. Those spins are natural candidates to model two level systems (TLS). The behavior of the cavity fields distribution for spins having stiffness close to the threshold one determines the tail of the low frequency density of states which is gapless. Therefore the corresponding spins are the natural candidates to model quasi localized modes (QLM) in glasses.
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