Existence of conference graphs for admissible orders

Determine which integers n congruent to 1 modulo 4 admit a strongly regular conference graph with parameters srg(n, (n−1)/2, (n−5)/4, (n−1)/4).

Background

The paper defines a conference graph as a strongly regular graph whose parameters are srg(n, (n−1)/2, (n−5)/4, (n−1)/4), where n is at least 5 and congruent to 1 modulo 4. Proposition 3.5 derives these parameter restrictions in the relevant eigenvalue case.

Although the parameters are arithmetically constrained, the existence of a corresponding conference graph is not known for every such order. The unresolved problem is to identify precisely which admissible values of n occur.