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Does Morley's congruence characterize primes greater than $3$?

In 1895 Morley showed that $$\binom{p-1}{(p-1)/2}\equiv(-1)^{\frac{p-1}2}4^{p-1}\pmod{p^3}$$ for any prime $p>3$. In 2009, I formulated the following conjecture concerning the converse of Morley's ...
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