Minimum value of l1 norm compatible with few even moments

I would like to know if this problem has already been studied in the literature. Let $r \in \mathbb{R}^m$ be a non-negative vector. I want to compute the optimization problem
\begin{equation} \label{eq:alpha-minimization} \alpha(r) = \operatorname{min}_{n \in \mathbb{N}}\operatorname{min}_{s \in [0,1]^n}\{\sum^{n}_{i=1} s_i \mid \forall k \in [m], \sum^n_{i=1} s^{2k}_i= r_k\} \end{equation}

in terms of $r$. Numerical evidence suggests that the minimum is obtained on $m+1$ distinct values, but I couldn't find a result showing this. Any references will also be really appreciated.