Does measurement with unknown outcome transform a pure superposition state into a mixed state?

$$\def\+{\!\!\kern0.08333em}$$ Say I have a spin-1/2 particle in a general, pure superposition state $$|\psi\rangle=\alpha|\+\uparrow\rangle+\beta|\+\downarrow\rangle,$$ or equivalently $$\rho=(\alpha|\+\uparrow\rangle+\beta|\+\downarrow\rangle)(\alpha^*\langle\uparrow\+|+\beta^*\langle\downarrow\+|)= \begin{bmatrix} |\alpha|^2 & \alpha\beta^*\\ \alpha^*\beta & |\beta|^2 \end{bmatrix}.$$ When I measure this state in the $$\{|\+\uparrow\rangle,|\+\downarrow\rangle\}$$ basis, I get $$|\+\uparrow\rangle$$ with probability $$|\alpha|^2$$ and $$|\+\downarrow\rangle$$ with probability $$|\beta|^2$$.

Now assume that this state was measured, but I do not know the outcome (e.g. my friend measured it but did not tell me the result, and will never do so). In this case, it seems appropriate, from my point of view, to describe the post measurement state as the mixed state $$\rho=|\alpha|^2|\+\uparrow\rangle\langle\uparrow\+|+|\beta|^2|\+\downarrow\rangle\langle\downarrow\+|$$. After all, I know that the measurement yielded either $$|\+\uparrow\rangle\langle\uparrow\+|$$ or $$|\+\downarrow\rangle\langle\downarrow\+|$$ with the given probabilities.

Is this reasoning correct? If no, why not?