Convergence of matrix inversion for divisor-function values

Let $M,N$ be integers and let $\mathbf{Q}_N$ be the matrix defined from the normalized functions $Q_k(2\pi n)$ by

$$\mathbf{Q}_N=\left\{Q_k(2\pi n)e^{-2\pi n}\right\}_{\substack{0\leqslant k\leqslant N-2\\1<n\leqslant N}}.$$

Let $\mathbf{1}$ be the all-ones vector.

Matrix convergence conjecture. The first $M-1$ entries of

$$-\mathbf{Q}_N^{-1}\mathbf{1}$$

tend to the values $\sigma_1(n)$ for $2\leqslant n\leqslant M$ as $N\to\infty$.

This gives a proposed deterministic recovery method for divisor-function values from the exponential-sum constraint system. The source does not provide a proof of the limiting assertion.