  In this paper we deal with the Cauchy problem for the hypodissipative
Navier-Stokes equations in the three-dimensional periodic setting. For all
Laplacian exponents $\theta<\frac13$, we prove non-uniqueness of dissipative
$L^2_tH^\theta_x$ weak solutions for an $L^2$-dense set of $\mathcal C^\beta$
H\"older continuous wild initial data with $\theta<\beta<\frac13$. This
improves previous results of non-uniqueness for infinitely many wild initial
data ([8,20]) and generalizes previous results on density of wild initial data
obtained for the Euler equations ([14, 13]).
