Adelic Dirac–Hilbert–Pólya conjecture for the Riemann zeros

Let $(\mathcal{D}_{\mathrm{arith}},\{\eta_p\})$ be a chiral adelic Dirac system, where the coefficient functions $\eta_p$ are even, and let $\xi_{\mathrm{Dirac}}$ denote its spectral shift function. The nontrivial zeros of $\zeta(s)$ have imaginary parts $\gamma_k$. Adelic Dirac–Hilbert–Pólya conjecture. There exist even coefficient functions $\eta_p$ satisfying

$$\sum_p\|\eta_p\|_\infty<\infty$$

such that $\xi_{\mathrm{Dirac}}$, after affine rescaling, reproduces the signed zero-counting measure

$$\sum_k\bigl(\delta(\lambda-\gamma_k)-\delta(\lambda+\gamma_k)\bigr)$$

for the nontrivial zeros of $\zeta(s)$. This is a chiral adelic analogue of the Hilbert–Pólya idea: the zeros are encoded by jump discontinuities of a spectral shift function rather than appearing directly as the spectrum of a self-adjoint Hamiltonian. The supplied text gives no resolution, so the conjecture remains open.