  Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every
$4$-connected planar triangulation $G$ on $n$ vertices has at least
$2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double
wheel. In this paper, we show that every $4$-connected planar triangulation on
$n$ vertices has $\Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if
$G$ is a $4$-connected planar triangulation on $n$ vertices and the distance
between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has
$2^{\Omega(n^{1/4})}$ Hamiltonian cycles.
