The dimension conjecture for spaces of real places of rational function fields

Let $K\subset\mathbb R$ be a subfield and let $n$ be a natural number. Write $M(K(x_1,\dots,x_n))$ for the space of $\mathbb R$-places of the rational function field $K(x_1,\dots,x_n)$, and let $\dim$ denote its topological dimension. A field $K$ is totally Archimedean if it is orderable and every total ordering on $K$ is Archimedean.

Dimension conjecture. For every such $K$ and $n$,

$$\dim M(K(x_1,\dots,x_n))\ge n.$$

If $K$ is totally Archimedean, then

$$\dim M(K(x_1,\dots,x_n))=n.$$

The statement concerns the relationship between algebraic properties of the coefficient field and the topological dimension of its space of $\mathbb R$-places. The supplied text gives no resolution status beyond the assertion itself, so the conjecture is recorded as open.