Sander's conjecture on rational points of Erdos-Selfridge superelliptic curves

For integers $k\geq 2$ and $\ell\geq 2$, consider the superelliptic curve

$$y^\ell=\frac{(x+1)^k-x^k}{2^k-1}.$$

The rational points include the trivial points with $y=0$ and the families

$$(x,y,k,\ell)=\left(\frac{a^2}{b^2-a^2},\frac{ab}{b^2-a^2},2,2\right),\qquad a\neq\pm b,$$ $$(x,y,k,\ell)=\left(\frac{1-2j}{2},\frac{\pm1}{2^j}\prod_{i=1}^j(2i-1),2j,2\right),$$

for integers $a,b$ and positive integer $j$, together with $(-4/3,2/3,3,3)$ and $(-2/3,-2/3,3,3)$. Sander's conjecture. If $k\geq2$ and $\ell\geq2$, then the only rational points on this curve satisfy $y=0$, or are among these displayed families for suitable choices of $a$, $b$ and $j$. This conjecture asserts that, in particular, there are no other rational points with $\ell\geq4$. The problem concerns effective classification of rational points on superelliptic curves; Faltings' theorem gives finiteness for each fixed curve of genus greater than one, but does not provide this uniform classification. The stated conjecture is attributed to Sander, with corrections noted by Bugeaud, Bugeaud, Győry and Mignotte.