Li-Yau gradient estimates on Riemannian manifolds with decaying Ricci curvature

Question

I am trying to understand a statement made in the paper "On gradient estimates for the heat kernel" by B Devyver below Proposition 1.26. The Li-Yau local gradient estimate says that ; for any $a > 1$ and $x \in B_R(x_0)$ and $t>0$ $$ \frac{|\nabla u|}{u^2} - \frac{a u_t}{u} \leq CR^{-2}a^2 (\frac{a^2}{a^2-1} + \sqrt{K}R) + \frac{na^2K}{2(a-1)} + \frac{na^2}{2t} $$ for some C>0 depending only on C, where $u$ is a solution of a heat equation on a riemannian manifold $M$ and $-K<0$ is the lower bound on the Ricci curvature on that ball. Below Proposition 1.26 in this paper it is said that this implies that for Riemannian manifolds where the Ricci curvature is decaying quadratically it actually holds $$ a \frac{|\nabla u|}{u^2} - \frac{ u_t}{u} \leq b (1/t+1/R^2) $$ for some positive constants $a,b$. I don't understand how to get rid of the constant factor $$ \frac{na^2K}{2(a-1)} $$ because the proof uses a maximum principle and if the maximum is attained in a point very near $x_0$ it isn't affected by the decay of the Ricci curvature.