  In 2021, Daqing Wan and Ping Xi studied the equivalence of the Lang-Trotter
conjecture for CM elliptic curves and the Hardy-Littlewood conjecture for
primes represented by a quadratic polynomial. Wan and Xi provided an
alternative description of the Lang-Trotter conjecture under the
Hardy-Littlewood conjecture. They obtained an explicit constant
$\overline{\omega}_{E,r}$ in the asymptotics of the Lang-Trotter conjecture.
They further conjectured that this particular constant would be equal to the
constant $C_{E,r}$ in the asymptotics of the original Lang-Trotter conjecture.
In this paper, we verify the same for $20$ CM elliptic curves, which also
establishes the equivalence of the Lang-Trotter Conjecture and the
Hardy-Littlewood Conjecture with respect to $r$, for these CM elliptic curves.
