Positive weights at the minimal exact discretization

Let $\Omega\subset\mathbb{R}^d$ be a compact subset, let $\mu$ be a finite measure on $\Omega$, and let $X_N$ be an $N$-dimensional subspace of the real space $L_2(\Omega,\mu)$ whose functions are defined at every point of $\Omega$. Write

$$m(X_N,w):=\min\{m:X_N\in\mathcal{M}^w(m,2,0)\}.$$

Suppose that nodes $\{\xi^j\}_{j=1}^m\subset\Omega$ and weights $\{\lambda_j\}_{j=1}^m$, with $m=m(X_N,w)$, satisfy

$$\int_\Omega f^2\,d\mu=\sum_{j=1}^m\lambda_j f^2(\xi^j)$$

for every $f\in X_N$. Positive-weight discretization conjecture. Then $\lambda_j>0$ for every $j=1,\dots,m$. This conjecture concerns whether a minimal exact weighted Marcinkiewicz-type discretization can always be chosen with, or must necessarily have, positive weights; the source formulates it as a conjecture posed in earlier work, and the supplied material gives no resolution.