Conjecture on signed measures with prescribed vanishing moments

Let $a,b\in \mathbb{R}$ with $0<a<b$. Let $(n_k)_{k \in \mathbb{N}}$ be a strictly increasing sequence of positive integers such that

$$\sum_{k=0}^\infty \frac{1}{n_k} < \infty.$$

A non-trivial signed measure with prescribed vanishing moments is a signed measure supported in $[a,b]$, absolutely continuous with respect to Lebesgue measure, and having a continuous density. Vanishing-moments conjecture. There exists such a measure for which all $n_k$th moments vanish. The conjecture would strengthen the preceding result, which guarantees infinitely many vanishing moments without controlling their particular exponents; the Müntz–Szász theorem motivates the reciprocal-summability condition.