based on the expression of the estimator in Eq. 54. This is the focused initial conditions proposed in the recently developed spin-LSC approach. However, these focused conditions do not provide any specific choice of {ϕ j } in generalized Euler angle variables (see Eq. 18), and based on the expression of dΩ (Eq. 21) it should be uniformly sampled. In principle, any algorithm that uses {q n , p n } should generate a uniform distribution of {ϕ j } in the range [0, 2π], required by the dΩ integral (see expression of dΩ in Eq. 21). In the spin-LSC approach, it is proposed that the corresponding angle variable φ n (Eq. 122b) should be sampled uniformly, as the original MMST formalism suggested. We shall see that ϕ n = φ n+1 -φ n (Eq. 125), thus randomly sample φ n is equivalent to randomly sample ϕ n . Our rigorous theoretical framework thus helps to justify the empirical choices made by the previous simulation method.