 Bessel power local martingale

For a Bessel process of dimension $\delta$, started at $x>0$, the Itô formula giveson its stochastic interval before hitting zero. Thus the power is a local martingale after localization away from zero. For $\delta \ge 2$ this interval is the whole lifetime; at $\delta =2$ the power is constant. The unqualified assertion fails for a reflecting Bessel process of dimension one, which acquires local time of a semimartingale at zero.