  We study instances of Beilinson-Tate conjectures for automorphic
representations of $\mathrm{PGSp}_6$ whose Spin $L$-function has a pole at
$s=1$. We construct algebraic cycles of codimension three in the Siegel-Shimura
variety of dimension six and we relate its regulator to the residue at $s=1$ of
the $L$-function of certain cuspidal forms of $\mathrm{PGSp}_6$. Using the
exceptional theta correspondence between the split group of type $G_2$ and
$\mathrm{PGSp}_6$ and assuming the non-vanishing of a certain archimedean
integral, this allows us to confirm a conjecture of Gross and Savin on rank $7$
motives of type $G_2$.
