Bounded-solution conjecture for generalized Ramanujan–Nagell equations

Let $g(n)$ be an integer-coefficient polynomial, and consider equation (5), namely

$$2^n\pm g(n)=T_m=\frac{(m+1)m}{2},\qquad n>2, m\geq 0, n,m\in\mathbf{Z}.$$

Bounded-solution conjecture. Equation (5) has at most $5$ positive integer solutions, where $D$ is an integer-coefficient polynomial of $n$.

The displayed conjecture occurs in a part of the source enclosed by a false-condition block and uses inconsistent notation, referring to $D$ after introducing $g$. The source gives no resolution status.