Isett's generic energy-profile regularity conjecture for Hölder Euler flows

Let $\alpha<1/3$. Consider weak periodic solutions of the incompressible Euler equations in the class $v\in C_tC_x^\alpha(\mathbb{R}\times\mathbb{T}^n)$, and define their energy profile by

$$e(t)=\int_{\mathbb{T}^n}|v|^2(t,x)\,dx.$$

Generic energy-profile regularity conjecture. A generic solution with regularity at most $1/3$ fails to conserve energy and has an energy profile of minimal regularity. More precisely, for every $\alpha<1/3$, there exists a solution such that

$$e\notin W^{2\alpha/(1-\alpha)+\epsilon,p}(I)$$

for every $\epsilon>0$, $p\geq1$, and every open interval $I\subseteq\mathbb{R}$; moreover, the set of all such solutions is residual in the space of weak solutions in $C_tC_x^\alpha(\mathbb{R}\times\mathbb{T}^n)$ endowed with the topology induced by the $C_tC_x^\alpha$ norm. This strengthens Onsager's conjecture by prescribing the sharp failure of energy-profile regularity; the source presents it as a main concern, and the parser supplies no evidence of resolution.