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⁾)ₙ, 1≤ j ≤ d, in the same random environment ω in Z, and a recurrent simple random walk (Zₙ)ₙ on Z. We assume that, conditionally on the environment ω, 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 ω is transient to the right but subballistic, with parameter 0<κ<1/2. We show that - for every value of d - there are almost surely infinitely many times for which all these random walks, (Zₙ)ₙ and (Sₙ⁽ʲ⁾)ₙ, 1≤ j ≤ d, are simultaneously at the same location, even though one of them is recurrent and the d others ones are transient.