This paper is a further investigation of the problem studied in [Xue & Zhao, Stochastic processes and their applications, 2020], where the authors proved a law of large numbers for the empirical measure of the weakly asymmetric normalized binary contact path process on Zd , d ≥ 3, and then conjectured that a central limit theorem should hold under a non-equilibrium initial condition. We prove that the aforesaid conjecture is true when the dimension d of the underlying lattice and the infection rate λ of the process are sufficiently large.