Distance bound for homogeneous polynomials with roots on the unit circle

Let $P$ be a homogeneous polynomial in two variables with $2r>d$ roots on the unit circle, and let $\Delta$ be the real discriminant. Write

$$\alpha=\dist(P,\Delta).$$

For the model polynomial $T_{n,d}$, let $\\|T_{n,d}\\|$ denote its Bombieri norm.

Distance-to-discriminant conjecture. One has

$$\alpha\leq\frac{\\|T_{n,d}\\|}{\\|P\\|}.$$

This is proposed after proving local maximality results for regularly spaced roots. Its status is not resolved in the supplied text.