Largest square divisors of shifted primes

The author shows that there are infinitely many primes $p$ such that for any nonzero integer $a$, $p-a$ is divisible by a square $d^2 > p^{\frac{1}{2}+\frac{1}{700}}$. The exponent $\frac{1}{2}+\frac{1}{700}$ improves Merikoski's $\frac{1}{2}+\frac{1}{2000}$. Many powerful devices in Harman's sieve are used for this improvement.