  This paper deals with the fully parabolic attraction-repulsion chemotaxis
system with signal-dependent sensitivities, \begin{align*} \begin{cases}
  u_t=\Delta u-\nabla \cdot (u\chi(v)\nabla v)
  +\nabla \cdot (u\xi(w)\nabla w),
  &x \in \Omega,\ t>0,\\[1.05mm]
  v_t=\Delta v-v+u,
  &x \in \Omega,\ t>0,\\[1.05mm]
  w_t=\Delta w-w+u,
  &x \in \Omega,\ t>0 \end{cases} \end{align*} under homogeneous Neumann
boundary conditions and initial conditions, where $\Omega \subset \mathbb{R}^n$
$(n \ge 2)$ is a bounded domain with smooth boundary, $\chi, \xi$ are functions
satisfying some conditions. Global existence and boundedness of classical
solutions to the system with logistic source have already been obtained by
taking advantage of the effect of logistic dampening (J. Math. Anal. Appl.;
2020;489;124153). This paper establishes existence of global bounded classical
solutions despite the loss of logistic dampening.
