The three-solution conjecture for $x^2+k^n=B$

Let $k\in\mathbb{Z}_{\geq 2}$ and let $B$ be a positive integer. A solution is a pair $(x,n)$ of non-negative integers satisfying

$$x^2+k^n=B.$$

Three-solution conjecture. For any such $k$ and $B$, the equation has at most $3$ solutions in non-negative integers $(x,n)$. The conjecture is motivated by numerical experiments for general $k$; the surrounding text exhibits equations with three solutions and gives no proof of the asserted universal upper bound.