The rational-root conjecture for Stern polynomials

Let $B_n(t)$ denote the Stern polynomial associated with the positive integer $n$. A rational number $a$ is a zero of $B_n(t)$ if $B_n(a)=0$ for some positive integer $n$. The rational-root conjecture. If $a\in\mathbb{Q}$ and there exists a positive integer $n$ such that $B_n(a)=0$, then

$$a\in\\{-1,-1/2,-1/3,0\\}.$$

This conjecture asserts that the displayed four values exhaust the rational zeros occurring among Stern polynomials; the supplied text gives no resolution beyond numerical and related computational evidence.