Definition: A poset P is $\mathbf{stationary}$ $\kappa^{+}$-$\mathbf{linked}$ if for every sequence of conditions $(p_{\gamma} | \gamma < \kappa^{+})$, there is a regerssive function $f: \kappa^{+} \to \kappa^{+}$ (i.e. $f(\gamma) < \gamma$, for $\gamma \neq 0$), such that for some club $C \subseteq \kappa^{+}$ and for all $\gamma, \delta \in C$ with cofinality $\kappa$, $f(\gamma) = f(\delta)$ implies $p_{\gamma}, p_{\delta}$ are compatible.
Construction of the poset: Let V be a model of $SMP_{2}(\omega_{\alpha + 1})$ ($\alpha$ is some fixed countable ordinal). That implies (We do not need the exact definition of SMP, it just gives us the next statements):
$GMA_{\omega_{\alpha + 1}}$ (I will not give definition for $GMA_{\omega_{\alpha + 1}}$, since we do not need it. In fact we need to prove the stationary $\aleph_{\alpha + 2}$-linked in order to use GMA)
$2^{\aleph_{\alpha}} = \aleph_{\alpha + 1}$
$2^{\aleph_{\alpha + 1}}$ is weakly inaccessible
Now let $\aleph_{\alpha + 2} \leq \lambda < 2^{\aleph_{\alpha + 1}}$ and P be the poset consists of conditions of the form $(t, f)$, where
$t$ is a tree of height $\beta + 1$ for some $\beta < \omega_{\alpha + 1}$ and levels at most of size $\aleph_{\alpha}$
$f$ is a function with $dom(f) \subseteq \lambda$, $|dom(f)| = \aleph_{\alpha}$ and $ran(f) = t_{\beta}$, where $t_{\beta}$ is the $\beta$th level of the tree $t$.
The order is defined as follows: $(u, g) \leq (t, f)$ iff
$t$ is an initial segment of $u$, $dom(f) \subseteq dom(g)$,
for every $\delta \in dom(f)$, either $f(\delta) = g(\delta)$ (if $t$ and $u$ have the same height) or $f(\delta) <_{u} g(\delta)$ otherwise.
This is a forcing notion that adds an $\aleph_{\alpha + 1}$-Kurepa tree with $\lambda$ cofinal branches.
I have already proved that it is $< \aleph_{\alpha + 1}$-closed and has the $\aleph_{\alpha + 2}$ chain condition. In order to prove the $\aleph_{\alpha + 2}$ chain condition I noticed that we have only $2^{\aleph_{\alpha}} = \aleph_{\alpha + 1}$-many possibilities for the first coordinate and then applied a $\Delta$-system Lemma on the domains of the second coordinate.
Question: Is P stationary $\aleph_{\alpha + 2}$-linked?
There is a Theorem for the case that $\alpha = 0$, which claims that poset is stationary $\aleph_{2}$-linked, but tells nothing more in the proof. So, I believe the answer is yes, but I cannot see how to prove it
