Continuity and piecewise real-analyticity of the infimum function

Let $\mathfrak{I}_{\mathcal{G}}(\sigma)$ be the infimum function associated with the optimization problem in the paper, where $\sigma$ is the parameter.

Continuity and real-analyticity conjecture. The function $\mathfrak{I}_{\mathcal{G}}(\sigma)$ is continuous in $\sigma$ and is real-analytic except at integers and half-integers.

The paper proves that this infimum function is strictly decreasing and proposes the stated regularity as a direction for future work. Its continuity and real-analyticity away from the specified exceptional points remain open in the source.