Question: Is option threadsafe heritable?

Suppose that a procedure is declared with option threadsafe and it has a local child procedure PC (possibly anonymous). Is their any benefit, or perhaps any detriment, to also declaring PC with option threadsafe? For example, is there any benefit or detriment to the yellow option threadsafe in this code below?:

P:= proc()
option threadsafe;
local PC:= proc()
option threadsafe; (* some code *) end proc;
    (* some code *)
    (* some code *)

end proc;

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