Separation at FL350 (35,000 feet) is a minimum of 1,000 feet vertical.
I should have clarified my earlier comments. What I meant was no ATC involvement would be needed if the airplane had an imminent issue where time was of the essence. This is a particularly poor area to fly through. No radar coverage, inter-tropical convergence zone, not a lot of other traffic to "help" you.
On Rush Limbaugh today, a pilot who flies this model Airbus speculated that the airplane lost both comm and radar and had to pick their way through the storms blind, and possibly went right to the heart of the beast instead of avoiding it.