The noncongruence of powers of two

Let $n$ be a positive integer power of $2$, so that $n\in 2^{\mathbb{N}}$. A positive integer is congruent if it is the area of a right triangle with rational side lengths. Noncongruence conjecture. If $n\in 2^{\mathbb{N}}$, then $n$ is not congruent. The claim concerns the congruent-number problem and asserts that no power of two occurs as the area of a rational right triangle. The supplied text gives no evidence that this claim has been proved or disproved.