Persistence of singular-hyperbolic large deviations for $C^1$ flows

Consider the statements of Theorem~ and the sectional-hyperbolic attracting-set conjecture for $C^2$ flows. $C^1$ regularity conjecture. The statements of Theorem~ and Conjecture~ should remain valid for $C^1$ flows. The source raises this as a regularity question and gives no resolution; in particular, the conjectured extension to $C^1$ flows remains open here.