Binomial-sum primality criterion for numbers of the form 4m plus or minus 1

Let $m$ be a positive integer. The binomial-sum primality conjecture. If $m\neq30$ and

$$\sum_{k=0}^{4m-2}{2k\choose k}m^k\equiv4m-1\pmod{(4m-1)^2},$$

then $4m-1$ is prime. If

$$\sum_{k=0}^{4m}{2k\choose k}(-m)^k\equiv4m+1\pmod{(4m+1)^2},$$

then $4m+1$ is prime. The authors report checking $m\leq1500$ without finding counterexamples, while presenting this as a sufficient condition for primality.