Tightness for the interface of the one-dimensional contact process

We consider a symmetric, finite-range contact process with two types of infection; both have the same (supercritical) infection rate and heal at rate 1, but sites infected by Infection I are immune to Infection 2. We take the initial configuration where sites in (-infinity, 0] have Infection I and sites in [1, infinity) have Infection 2, then consider the process rho(t) defined as the size of the interface area between the two infections at time t. We show that the distribution of rho(t) is tight, thus proving a conjecture posed by Cox and Durrett in [Bernoulli 1 (1995) 343-370].