Infinitely many collisions between a recurrent simple random walk and arbitrary many transient random walks in a subballistic random environment

We consider d random walks (Sₙ^ (j) ) ₍, 1 j d, in the same random environment in Z, and a recurrent simple random walk (Zₙ) ₍ on Z. We assume that, conditionally on the environment, all the random walks are independent and start from even initial locations. Our assumption on the law of the environment is such that a single random walk in the environment is transient to the right but subballistic, with parameter 0<<1/2. We show that - for every value of d - there are almost surely infinitely many times for which all these random walks, (Zₙ) ₍ and (Sₙ^ (j) ) ₍, 1 j d, are simultaneously at the same location, even though one of them is recurrent and the d others ones are transient.