Inverse powers of Bessel process not a martingale

Question

I am trying to show that $X_t^{1-2a}$ is not a martingale, where the solution $X_t$ to $$dX_t = \frac{a}{X_t} \ dt + \ dB_t, \ \ X_0=1$$ for $a>1/2$. I am able to do this directly for $a = (d-1)/2$, because then $X_t$ is the norm of the standard $d-$dimensional Brownian motion starting on the sphere of radius $1$. I also have the following facts: $X_t \to \infty$ almost surely as $t \to \infty$, as well as $$\mathbb{P}(X_t \text{ hits } R \text{ before } \epsilon) = \frac{1- \epsilon^{1-2a}}{R^{1-2a} - \epsilon^{1-2a}}.$$ Here $\epsilon < 1<R$. If $X_t$ had started at $x$ we would replace $1$ there with $x^{1-2a}$. But I am not sure how to use these facts to show that $M_t\equiv X_t^{1-2a}$ is not a martingale. I can't seem to justify the limit swap to show $\mathbb{E}[M_t] \to 0$. In particular there doesn't seem to be anything I can dominate with. If there were some uniform integrability I could use Egorov and be done, but I just don't know enough about the Bessel process to know if this is true. Does anyone have any hints?