## Is there a prime $q$ smaller than a given prime $p>5$ such that the inverse of $q$ modulo $p$ is an integer square?

Let $$p$$ be a prime. For each $$k=1,\ldots,p-1$$ there is a unique $$\bar k\in\{1,\ldots,p-1\}$$ with $$k\bar k\equiv1\pmod p$$, and we call $$\bar k$$ the inverse of $$k$$ modulo $$p$$. In 2014 I investigated the set $$\{\bar q:\ q\ \text{is a prime smaller than}\ p\}$$ and found that it contains an integer square if $$5. (See http://oeis.org/A242425 and http://oeis.org/A242441.) This led me to formulate the following conjecture.

Conjecture. For any prime $$p>5$$, there is a prime $$q such that the inverse $$\bar q$$ of $$q$$ modulo $$p$$ is an integer square.

For example, the inverse of $$13$$ modulo $$23$$ is $$4^2<23$$, the inverse of $$5$$ modulo $$61$$ is $$7^2<61$$, and the inverse of $$11$$ modulo the prime $$509$$ is $$18^2<509$$.

QUESTION. What tools in number theory are helpful to prove the above conjecture?