Existence of a third self-similar solution of the Yang--Mills equation in high odd dimensions

In a recent paper~\cite{BGW}, Bizoń, Glogić, and Wasserman (BGW) reduced the classification of smooth self-similar Yang--Mills profiles in odd spatial dimensions $d\ge11$ to an algebraic problem of counting roots of an explicit polynomial in a prescribed interval. Two roots correspond to explicit profiles. BGW conjectured that there is exactly one additional root and verified this in a range of odd dimensions. Here, we prove the existence of an additional root, and hence of a third smooth self-similar profile, in every odd dimension $d\geq11$. The proof combines an endpoint sign calculation for the resonance polynomial with an algebraic argument showing that the third root is distinct from the two explicit ones. Uniqueness remains open in general.