For Hilbert spaces $\mathcal{H}_X$, $\mathcal{H}_Y$ and $\mathcal{K}$, consider a linear map $f\colon \mathcal{K} \oplus \mathcal{H}_X \to \mathcal{K} \oplus \mathcal{H}_Y$ that is given as a matrix

$$f = \begin{pmatrix} A \colon \mathcal{K} \to \mathcal{K} & B\colon \mathcal{H}_X \to \mathcal{K} \\ C\colon \mathcal{K} \to \mathcal{H}_Y & D\colon \mathcal{H}_X \to \mathcal{H}_Y \end{pmatrix}.$$

Conjecture. Assume $f$ is nonexpansive (i.e. $\lvert\!\lvert f \rvert\!\rvert_{op} \leq 1$) and $A$ is strictly contractive (i.e. $\lvert\!\lvert A \rvert\!\rvert_{op} < 1$). Then, the map $$ \Uparrow\!(f) = D + C \sum^\infty_{n=0} A^n B $$ is nonexpansive.

Notice that $\Uparrow\!(f)$ is well-defined because $A$ is assumed strictly contractive and therefore the series converges.

I know the conjecture above is true when $f$ is an isometry; in that case $\Uparrow\!(f)$ turns out to be an isometry too. This is proven in page 24 from https://arxiv.org/pdf/1404.0074.pdf. It seems natural that if it's true for isometries it should hold for any nonexpansive map, however I have not been able to work out the technical details to prove it.

I would expect this to be a well-known result; I'd be grateful if you could reference some text where this is proven or give me a hint on how to proceed.