[see also: answer, explanation]
This equation has a unique solution for every $p$.
To guarantee existence and uniqueness of a solution to (3), the function $g$ must be ......
Fortunately, there is a very satisfactory solution to this problem, due to Vermes.
This observation prompted the author to look for a more constructive solution.
It seems that the solution of Problem 1 is still out of reach.
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