Centered subcritical Vaughan moment conjecture

Let $K_s(\chi)$ be the short coefficient-$1$ character polynomial, let $D_s(\chi)$ be the localized Möbius polynomial, and let $P_{s,L}^{\circ}(\chi)$ be the centered prime character sum. For dyadic parameters satisfying

$$K\le SX^\eta,\qquad DKL\asymp P,\qquad S^2\ll L,$$

Centered subcritical Vaughan moment conjecture. One has

$$\sum_{s\asymp S}\frac{1}{\varphi(s)}\sum_{\chi\ne\chi_0}|K_s(\chi)|^2|D_s(\chi)|^2|P_{s,L}^{\circ}(\chi)|^2\ll_\varepsilon SDKL\,X^\varepsilon.$$

The principal character and explicitly separated exceptional or major-arc modes are removed. This is the residual short-free-variable estimate left after the unconditional diagonal, geometric, energy, and long-line analyses; its status is the remaining unresolved spectral input.