The four-solution conjecture for a parametrized exponential Diophantine equation

Let $m$ be a positive integer. Consider the Diophantine equation

$$x^2+(2^{m+1}+1)2^n=2^{4(m+1)}+2^{3(m+1)}+2^{2m}+2^{m+1}+1.$$

The four-solution conjecture. This equation has exactly four solutions in integers $(x,n)$, with

$$n=0,\quad m+2,\quad 2m+3,\quad 3m+3.$$

The conjecture concerns the case $x^2=A2^n+B$ with $A<0$ and is motivated by numerical and structural experiments; no resolution is given in the supplied text.