Ratio of differences of fourth powers conjecture

A number $t$ is represented as a ratio of two differences of fourth powers if there exist integers or rational numbers $a,b,c,d$ such that

$$t=\frac{a^4-b^4}{c^4-d^4},$$

with $c^4-d^4\neq 0$. Ratio of differences of fourth powers conjecture. Every number can be written as the ratio of two differences of fourth powers. This conjecture arises from the study of rational points on diagonal quartic surfaces; the source notes that the corresponding density statement has been verified for parameters with numerator and denominator at most $100$, but does not provide a general proof.