The equioscillating-spline bound for quadrature weights

Let $\bm{x}$ be a set of unisolvent interpolation points, let $c(\bm{x})$ denote the equioscillating spline associated with these points, let $I(1)$ denote the integral of the constant function $1$, and let $\bm{w}$ be the associated quadrature-weight vector. Then, the equioscillating-spline bound asserts that

$$I(1) \leq \|\bm{w}\|_1 \leq \|c\|I(1).$$

This would improve the general condition-number bound for quadrature rules based on spline interpolation, particularly when negative weights occur; its validity is left as an open problem.