Suppose that $F$ is a totally real number field such that $[F:{\Bbb Q}]$ is odd. Then we shall choose the quaternion algebra $D$ everywhere unramified at finite places and at all but one infinite place. That is, $D$ splits at exactly one infinite (real) place, which is the so-called Eichler condition.
Then, ${\mathrm{GL}}(2)$ and $D_A ^\times$ both have the same finite part; so, they have equal $\Gamma_0(N)$ groups. The intersection $D \cap \Gamma_0(N)$, which is our Eichler order, seems to be well-defined up to the conjugacy of $\Gamma_0(N)$ in ${\mathrm{GL}}(2)$. The number of non-isomorphic Eichler orders in $D$ is defined as the type number $t_N \geq 1$, which can be truly bigger than one.
The Arithmetic of Quaternion Algebra by M-F Vigneras states that $h^{+} = {\mathrm{odd}}$ is the sufficient condition for $t_N$ to be $1$, where $h^{+}$ is defined to be the class number of ideas of $F$ in the strict sense induced by all of the classes real infinite places of $F$.
Question. When defining Shimura curve $C$ of level $N$ from the quaternion algebra $D$ over $F$, do experts very often take it granted that the above condition, i.e., $h^{+} = {\mathrm{odd}}$ holds for $F$?
For example does the totally real number field ${\Bbb Q}_n$, which is the $n$-the layer of the unique ${\Bbb Z}_p$-extension ${\Bbb Q}_{\infty}$ of ${\Bbb Q}$, satisfy that $h^{+} = {\mathrm{odd}}$ ?