I was wondering whether, and, if yes, how, weak convergence w.r.t. the Skorokhod $J_1$ topology relates to u.c.p. convergence. More specifically, if we have u.c.p. convergence, does this imply weak convergence w.r.t. the Skorokhod $J_1$ topology? For the other direction, I know that Skorokhod convergence does not imply uniform convergence; however, does the same hold for u.c.p. convergence?
Of course those questions assume a stochastic processes context. I'm willing to specify this to point processes, if necessary.
I tried looking this up online and in a few textbooks on convergence of stochastic processes, but couldn't find any answers. Any help is much appreciated.