Beukers' conjecture on solutions of the generalized Lebesgue–Ramanujan–Nagell equation

Let $D$ be a fixed positive nonsquare integer, let $p$ be a fixed prime with $\gcd(D,p)=1$, and let $N(-D,2)$ denote the number of solutions $(x,n)$ in $\mathbb{N}^2$ of

$$x^2-D=2^{n+2}.$$

The exceptional conditions $(i)$, $(ii)$, and $(iii)$ are the three conditions on $D$ listed above in the source. Beukers' conjecture.

$$N(-D,2)\leq 2,$$

except when $D$ satisfies one of the conditions $(i)$, $(ii)$, or $(iii)$. This conjecture concerns the sharpened bound for the number of solutions after the known result $N(-D,2)\leq 3$, apart from condition $(i)$; its status is not specified in the source.