Convergence of SVS-SPRING for linear least-quadratics

Let $J$ define a consistent instance of linear least-quadratics, let $\theta^*$ be a solution, and assume an initial guess $\theta_0\in\operatorname{range}(J^\top)$. Let SVS$(J,k,\lambda)$ denote the sampling distribution, and let $\alpha$ be as in, $\beta$ as in, and $\gamma$ as in the stated LLQ theorem. With appropriate choices of $\eta$ and $\mu$, SPRING is expected to satisfy

$$\mathbb{E}\,\left\lVert \theta_t-\theta^*\right\rVert^2=\mathcal{O}\left(\left(1-\kappa^{-1}(H)\cdot\sqrt{\alpha/\beta}\;/\;\gamma\right)^t\right)\left\lVert\theta_0-\theta^*\right\rVert^2.$$

Convergence of SVS-SPRING for LLQ. Under these assumptions and the SVS$(J,k,\lambda)$ sampling distribution, SPRING should satisfy the displayed convergence bound, where the hidden constant in $\mathcal{O}$ does not depend on $t$. The conjecture proposes that SPRING accelerates SNG in the linear least-quadratics setting by replacing $\alpha$ with the accelerated sketch-and-project factor $\sqrt{\alpha/\beta}$, modified by $\kappa^{-1}(H)/\gamma$. The paper states that the result is not currently proved because an appropriate Lyapunov function combining Nesterov acceleration with the nontrivial function-space Hessian is unclear.