Unrestricted-support conjecture for the low-lying zeros theorem

Let $\phi$ be a suitable test function, and let Theorem denote the density theorem established in the paper for the family of elliptic curves. Unrestricted-support conjecture. Theorem holds for test functions $\phi$ with no restrictions on the support. This would extend the low-lying-zero density result beyond the support currently accessible by the authors' methods, potentially distinguishing the orthogonal symmetry types $O$, $SO(\operatorname{even})$, and $SO(\operatorname{odd})$.