The following is from:

C.L. Stewart, "On the number of solutions of polynomial congruences and Thue equations", Journal of the American Math. Soc., 4 (1991), 793-835.

"Bombieri and Schmidt showed (in E. Bombieri and W.M. Schmidt, "On Thue's Equation", Invent. Math, 88., (1987), 69-81) that $F(x,y) = h$ where $F(x,y) = a_r x^r + a_{r-1}x^{r-1}y + \cdots + a_0 y^r$ may have at least $r$ distinct primitive solutions (a solution $(x,y)$ is primitive if $\gcd(x,y) = 1$). They gave the example $$\displaystyle F(x,y) = x^r + a(x-y)(2x-y)\cdots(rx -y),$$ where $a$ is a non-zero integer. Then $(1,1), (1,2), \cdots, (1,r)$ are primitive solutions of $F(x,y) = 1$. We do not believe that for a fixed form $F$ there are infinitely many integers $h$ for which $F(x,y) = h$ has this many primitive solutions if $r$ is large. Indeed we conjecture that there exists an absolute constant $c_0$ such that for any binary form $F \in \mathbb{Z}[x,y]$ with non-zero discriminant and degree at least three there exists a number $C$, which depends on $F$, such that if $h$ is larger than $C$ then the Thue equation $F(x,y) = h$ has at most $c_0$ solutions in coprime integers $x$ and $y$.

I am wondering what progress, if any, there is on this conjecture.