Neumann Li-Yau gradient estimate under integral Ricci curvature bounds

We prove a Li-Yau gradient estimate for positive solutions to the heat equation, with Neumann boundary conditions, on a compact Riemannian submanifold with boundary $\textbf {M}^n\subseteq \textbf {N}^n$, satisfying the integral Ricci curvature assumption: \begin{equation} D^2 \sup _{x\in \textbf {N}} \left ( \oint _{B(x,D)} |Ric^-|^p dy \right )^{\frac {1}{p}} < K \end{equation} for $K(n,p)$ small enough, $p>n/2$, and $diam(\textbf {M})\leq D$. The boundary of $\textbf {M}$ is not necessarily convex, but it needs to satisfy the interior rolling $R-$ball condition.