Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda \to \mathbb{R}$ be a continuous function for which we know the following: for all $x\in\Lambda$, it holds that ${\bf E}( h(X_{\tau_r})\mid X_0=x) = h(x)$ for all small enough $r>0$, where $\tau_r$ is the hitting time of the circumference of radius $r$ centred at $x$.

Can one conclude from this that $h(X_t)$ is a local martingale?

Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda \to \mathbb{R}$ be a continuous function for which we know the following: for all $x\in\Lambda$, it holds that ${\bf E}( h(X_{\tau_r})\mid X_0=x) = h(x)$ for all small enough $r>0$, where $\tau_r$ is the hitting time of the circumference of radius $r$ centred at $x$.

Can one conclude from this that $h(X_t)$ is a local martingale?