The conjecture that the logarithmic loss in the general estimate is technical

Let $a3$ be smooth and parametrized by $b3$, with $0<|\gamma'|<c_{\max}$. For $s\in\mathbb R$, $N>0$, $k_0>0$, $k>k_0$, and $0<\epsilon<1$, let $F_{\mathscr L}^{s,\epsilon}(N,L)$ denote the quantity defined in the paper.

The technical-loss conjecture. For every $k_0>0$ and $s\in\mathbb R$, there is $C>0$ such that, for all $N>0$, $k>k_0$, and $0<\epsilon<1$,

$$F_{\mathscr L}^{s,\epsilon}(N,L)\leq C.$$

The conjecture asserts that the $\sqrt{\log k}$ loss occurring in the currently available estimates is technical rather than intrinsic. The source indicates that the result is proved for strictly convex boundaries with non-vanishing curvature, while the general smooth case remains open.