  We prove genuinely sharp estimates for the Jacobi heat kernels introduced in
the context of the multidimensional cone $\mathbb{V}^{d+1}$ and its surface
$\mathbb{V}^{d+1}_0$. To do so, we combine the theory of Jacobi polynomials on
the cone explored by Xu with the recent techniques by Nowak, Sj\"ogren, and
Szarek, developed to find genuinely sharp estimates for the spherical heat
kernel.
