We investigate spherical 4-distance 7-designs by studying their distance distributions. We compute these distance distributions and use their product (an integer) to derive certain divisibility conditions relating the dimension $n$ and the cardinality $M$ of our designs. It follows that $n$ divides $12M$ and $n+1$ divides $4M^2$. This result provides a good base for computer experiments to support the folklore conjecture that the only spherical 4-distance 7-designs are the tight spherical 7-designs. We then proceed with a computer assisted proof of this conjecture in all dimensions $n \leq 1000$.