The cubic binary-form perfect-power conjecture

Let $F$ be a separable homogeneous cubic binary form with integer coefficients, let $d$ be a fixed integer with $d\geq 1$, and let $l$ be a prime number. Cubic binary-form perfect-power conjecture. There exists a constant $C_{d,F}>0$, depending only on $d$ and $F$, such that if $l>C_{d,F}$ and

$$F(a,b)=dc^l$$

with $\gcd(a,b)=1$, then $c=\pm1$. This conjecture is given as a sufficient statement from which the earlier large-exponent conjecture would follow; the paper does not establish it.