Generic strict inequality for physical measures and Lorenz-like singularities

Consider the family of $3$-vector fields $G$ of class $C^2$ exhibiting a singular-hyperbolic attracting set containing some Lorenz-like singularity. Write $s$ for the number of ergodic physical measures and $s_L$ for the relevant bound associated with Lorenz-like singularities. Genericity conjecture. There exists an open and dense subset of vector fields for which the bound $s<s_L$ holds for each singular-hyperbolic attracting set. The conjecture proposes that the strict bound is generic because the periodic-orbit configuration used in the sharp examples is not robust under small perturbations.