reduction of a conjugate point in $X_0(p)$

Consider a non-cuspidal point $x\in X_0(p)(K)$ representing an elliptic curve $E/K$ over a quadratic field $K$, and suppose that $E$ has potentially multiplicative reduction over a prime $\mathfrak{q}$ of $K$ over the rational prime $q$. Let $0,\infty$ be the cusps of $X_0(p)$.

Using the notation $x|_{\mathbb{F}_{\mathfrak{q}}}$ for mod $\mathfrak{q}$ reduction in $X_0(p)(K)$ we have

$$ x|_{\mathbb{F}_{\mathfrak{q}}} \in \big\{ 0|_{\mathbb{F}_{\mathfrak{q}}}\ ,\ \infty|_{\mathbb{F}_{\mathfrak{q}}} \big\} $$

Let $\tau$ generate $Gal(K/\mathbb{Q})$.

What can we say about $x^\tau|_{\mathbb{F}_{\mathfrak{q}}}$?

In particular, suppose $q$ doesnt split in $K$ so there is only one prime $\mathfrak{q}$ in $K$ over $q$. Then how can I prove that both $x$ and $x^\tau$ reduce to the same cuspidal point? i.e:

$$ (x,x^\tau)|_{\mathbb{F}_{\mathfrak{q}}} \in \big\{ (0,0)|_{\mathbb{F}_{\mathfrak{q}}}, \ (\infty,\infty)|_{\mathbb{F}_{\mathfrak{q}}} \big\} $$

Where the notation $(x,x^\tau)$ refers to a point in the symmetric curve $X_0^{(2)}(p)$.