The converse to stability implying p-stability for connected real symmetric pairs

Let $(G,H,\theta)$ be a symmetric pair over ${\mathbb{R}}$, where $G$ is connected in the real topology. The pair is p-stable if it satisfies the p-stability condition discussed in the paper.

Converse stability conjecture. If $(G,H,\theta)$ is p-stable, then $(G,H,\theta)$ is stable.

The preceding theorem proves that stability implies p-stability. The converse is presented as an empirical expectation in the Archimedean case, at least for connected $G$, and remains open based on the supplied source.