  For any regularity exponent $\beta<\frac 12$, we construct non-conservative
weak solutions to the 3D incompressible Euler equations in the class $C^0_t
(H^{\beta} \cap L^{\frac{1}{(1-2\beta)}})$. By interpolation, such solutions
belong to $C^0_tB^{s}_{3,\infty}$ for $s$ approaching $\frac 13$ as $\beta$
approaches $\frac 12$. Hence this result provides a new proof of the flexible
side of the $L^3$-based Onsager conjecture. Of equal importance is that the
intermittent nature of our solutions matches that of turbulent flows, which are
observed to possess an $L^2$-based regularity index exceeding $\frac 13$. Thus
our result does not imply, and is not implied by, the work of Isett [A proof of
Onsager's conjecture, Annals of Mathematics, 188(3):871, 2018], who gave a
proof of the H\"older-based Onsager conjecture. Our proof builds on the
authors' previous joint work with Buckmaster et al. (Intermittent convex
integration for the 3D Euler equations: (AMS-217), Princeton University Press,
2023), in which an intermittent convex integration scheme is developed for the
3D incompressible Euler equations. We employ a scheme with higher-order
Reynolds stresses, which are corrected via a combinatorial placement of
intermittent pipe flows of optimal relative intermittency.
