  We explore the thinness of hypergeometric groups of type $\mathrm{Sp}(4)$ and
$\mathrm{Sp}(6)$ by applying a new approach of computer-assisted ping pong. We
prove the thinness of $17$ hypergeometric groups with maximally unipotent
monodromy in $\mathrm{Sp}(6)$, completing the classification of all $40$ such
groups into arithmetic and thin cases.
  In addition, we establish the thinness of further $46$ hypergeometric groups
in $\mathrm{Sp}(6)$, and of $3$ hypergeometric groups in $\mathrm{Sp}(4)$,
completing the classification of all $\mathrm{Sp}(4)$ hypergeometric groups. To
the best of our knowledge, this article produces the first $63$ examples in the
cyclotomic family of Zariski dense non-arithmetic hypergeometric monodromy
groups of real rank three.
