  We study the non-uniqueness of weak solutions for the two-dimensional
hyper-dissipative Navier-Stokes equations in the super-critical spaces
$L_{t}^{\gamma}L_{x}^{p}$ when $\alpha\in[1,\frac{3}{2})$, and obtain the
conclusion that the non-uniqueness of the weak solutions at the endpoint
$(\gamma,p)=(\infty, \frac{2}{2\alpha-1})$ is sharp in view of the generalized
Lady\v{z}enskaja-Prodi-Serrin condition by using a different spatial-temporal
building block from [Cheskidov-Luo, Ann. PDE, 9:13 (2023)] and taking advantage
of the intermittency of the temporal concentrated function $g_{(k)}$ in an
almost optimal way. Our results recover the above 2D non-uniqueness conclusion
and extend to the hyper-dissipative case $\alpha \in(1,\frac{3}{2})$.
