Removing the logarithmic loss in the Li–Wu KdV integrator

Consider the real periodic KdV equation $$\partial_t u + \partial_x^3 u = \tfrac12 \partial_x(u^2), \qquad x \in \mathbb{T} = \mathbb{R}/(2\pi\mathbb{Z}), \qquad u(0) = u_0,$$ where $u_0 \in H^\gamma(\mathbb{T})$ has mean zero and $0 < \gamma \le 1$. Let $S_t = e^{-t\partial_x^3}$, $P_0 f = (2\pi)^{-1} \int_{\mathbb{T}} f$ and $P = I - P_0$. Let $D^{-1} = \partial_x^{-1}$ be the multiplier with symbol $(ik)^{-1}$ for $k \ne 0$ and $0$ for $k = 0$, and $D^{-2} = (D^{-1})^2$. For $T > 0$ and $\tau = T/L$, the Li–Wu time discretization is $u^0 = u_0$ and $$u^{n+1} = S_\tau u^n + F_\tau(u^n) + H_\tau(u^n),$$ where $$F_\tau(f) = \tfrac16 P\bigl[(S_\tau D^{-1} f)^2\bigr] - \tfrac16 S_\tau P\bigl[(D^{-1} f)^2\bigr]$$ and $$\begin{aligned} H_\tau(f) ={}& \tfrac13 P\bigl[(S_\tau D^{-1} f)\, D^{-1} F_\tau(f)\bigr] + \tfrac{\tau}{9} (S_\tau D^{-1} f)\, P_0(f^2) \\ &- \tfrac{1}{54} \Bigl[S_{\tau - s} D^{-1}\bigl[(S_s D^{-1} f)^3\bigr]\Bigr]_{s=0}^{s=\tau} \\ &- \tfrac{1}{27\tau} \Bigl[S_{\tau - s} D^{-2}\bigl[(S_{s - \tau} D^{-2} F_\tau(f))(S_s D^{-1} f)\bigr]\Bigr]_{s=0}^{s=\tau}, \end{aligned}$$ the brackets denoting the value at $s = \tau$ minus the value at $s = 0$. This is Li and Wu's scheme, without a spatial frequency cutoff. They proved that there are $\tau_0 > 0$ and $C$, depending only on $\|u_0\|_{H^\gamma}$, $\gamma$ and $T$, such that $$\max_{1 \le n \le L} \|u(n\tau) - u^n\|_{L^2(\mathbb{T})} \le C \tau^\gamma \ln(1/\tau) \qquad (\tau \le \tau_0),$$ and note that the logarithmic factor arises from technical estimates in their proof.

Problem. For every $T > 0$ and $0 < \gamma \le 1$, is $$\max_{0 \le n \le L} \|u(n\tau) - u^n\|_{L^2(\mathbb{T})} \le C \tau^\gamma$$ for all $\tau = T/L \le \tau_0$, with $C$ and $\tau_0 > 0$ depending only on $T$, $\gamma$ and $\|u_0\|_{H^\gamma}$? The question concerns this particular scheme, and no smoothness of $u_0$ beyond $H^\gamma$ is assumed.