Signed-measure converse conjecture for stationary distributions of diffusions

Let $X$ be a diffusion process on $\mathbb{R}^{d}$ with a unique stationary distribution, and let $\mathcal{G}$ be its generator. Suppose that a signed measure $\pi$ on $\mathbb{R}^{d}$ satisfies

$$\int_{\mathbb{R}^{d}}\mathcal{G}f(x)\,\pi(\mathrm{d}x)=0\qquad\text{for all }f\in C_{b}^{2}(\mathbb{R}^{d})$$

and $\pi(\mathbb{R}^{d})=1$.

Stronger converse conjecture. Then $\pi$ is a nonnegative measure and consequently it is a stationary distribution of $X$.

This would strengthen the converse theorem for probability measures satisfying the basic adjoint relationship by showing that the normalization and adjoint relationship already force positivity. The source gives no resolution.