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<p<2\times 10^8$ . (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<p$ 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?