Conjecture on strongly extreme points via a unique inner minimizer

Prove that for any proper weak-star closed subspace E of H^∞ and any unit-norm coset f+E in H^∞/E, the coset f+E is a strongly extreme point of the unit ball in H^∞/E if and only if the set M(f; H^∞, E) = { h ∈ f+E : ||h||_∞ = ||f+E|| } contains exactly one function and this function is inner (i.e., has boundary modulus 1 almost everywhere on the unit circle).

Background

Building on their description of extreme points in H^∞/E and the known fact that inner functions are precisely the strongly extreme points in H^∞, the authors propose a criterion for strongly extreme points in the quotient setting.

The conjecture formulates a concrete equivalence in terms of the uniqueness and inner nature of the norm-attaining representative within the coset, using the notation M(f; H^∞, E) introduced earlier for the set of points of minimum norm in a given coset.