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Variable assignment is not strictly necessary, what is necessary is that one loop iteration can modify the initial state of variables in the next loop iteration. This is the second alternative to setting the current value of a variable presented in Flow (specifically, recurrence relations, e.g. x' = x + 1, which are syntactic sugar but allow "variable assignment" but only allow assignment to variables accessible in the next loop iteration). So the Flow Manifesto doesn't cover the hole in the Böhm-Jacopini theorem specifically, but that doesn't mean this computational model is not Turing complete.

As far as proofs or working implementation, sometimes life gets in the way of progress :-)

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