The local-global conjecture for the fourth level of number fields

Let $K$ be a number field with finite fourth level, and let $s_4(K)$ denote the least number of fourth powers of elements of $K$ whose sum is $-1$ when such a representation exists. For each prime ideal $\mathfrak{p}$ dividing $(2)$, let $K_\mathfrak{p}$ be the corresponding $\mathfrak{p}$-adic completion, with fourth level $s_4(K_\mathfrak{p})$. Local-global conjecture for the fourth level. One has

$$s_4(K)=\max_{\mathfrak{p} \mid (2)}s_4(K_\mathfrak{p}).$$

The conjecture proposes that, whenever the fourth level is finite, it is determined by the fourth levels of the completions at the prime ideals above $2$. The paper notes that all cases where both the lower bounds and the global fourth level can be computed give equality, while the possible values of $s_4(K)$ for number fields remain incompletely understood.