Finite-supremum conjecture for polynomial generalized Ramanujan–Nagell equations

Let $k$ be a positive integer and let $D(n)$ be an integer-coefficient polynomial with $\deg(D)\leq k$. Consider the generalized Ramanujan–Nagell equation

$$x^2+D(n)=2^n.$$

Finite-supremum conjecture. The supremum of the number of solutions of this equation, as $D$ ranges over integer-coefficient polynomials with $\deg(D)\leq k$, exists; denote it by $N(k)$. Moreover,

$$N(k)\geq k.$$

The surrounding discussion constructs examples with arbitrarily many solutions when the degree is allowed to vary, motivating a degree-dependent extremal quantity. The source gives no resolution status.