Extremality of generalized Chebyshev forms

Determine whether the binary form C_{r,d}=(X_1^2+X_2^2)^{(d-r)/2}\operatorname{Re}(X_1+iX_2)^r maximizes the Bombieri–Weyl distance to the real discriminant among real binary degree-d forms having 2r real zeros and the same Bombieri–Weyl norm, for every d/3<r\le d with d-r even.

Background

The paper studies extremal problems for the Bombieri–Weyl distance from real homogeneous forms to the real discriminant. For binary forms, Raffalli proposed a family of candidate maximizers indexed by r, obtained by multiplying the real part of a complex power by a power of X_1^2+X_2^2. The authors prove the conjecture only in the real-rooted case r=d, where the candidate is the Chebyshev form C_{d,d}; the cases with r<d remain unresolved.