Folklore finiteness conjecture for bounded-degree isogenous K3 surfaces

Let $N$ be an integer and let $X$ be a K3 surface over $\overline{\mathbf{Q}}$. Consider the K3 surfaces isogenous to $X$ that can be defined over extensions of degree at most $N$. Folklore conjecture. The number of their isomorphism classes can be bounded solely in terms of $N$. This is a finiteness prediction for isogeny classes of K3 surfaces with bounded-degree fields of definition; the source states it as folklore and provides no resolution.