A problem about holomorphic functions not continuous to the boundary. I'm stuck in the the following argument. I believe the following to be true but not able to prove.
Let $\varphi_i\in\mathcal{O}(\mathbb{D},\mathbb{D}),\,i=1,\,2.$ Fix a point $p\in\partial\mathbb{D}.$ Suppose there exist sequences, $\{z_\nu^i\}_{\nu\in\mathbb{Z}_+},\,i=1,2,$ converging to $p$ such that the sequences, $\{\varphi_i(z_\nu^i)\}_{\nu\in\mathbb{Z}_+},\,i=1,2,$ converges in $\mathbb{D}.$ Furthermore the sequences $\{\varphi_1(z_\nu^2)\}_{\nu\in\mathbb{Z}_+}$ and $\{\varphi_2(z_\nu^1)\}_{\nu\in\mathbb{Z}_+}$ converges to the boundary of $\mathbb{D}.$ 
Claim: There exists a sequence $\{z_\nu\}_{\nu\in\mathbb{Z}_+}$which converges to $p$ such that $\{\varphi_i(z_\nu)\}_{\nu\in\mathbb{Z}_+},\,i=1,2$ converges in $\mathbb{D}.$
 A: 
Furthermore the sequences $\{\varphi_1(z_2^\nu)\}$ and $\{\varphi_2(z_1^\nu)\}$ converges to the boundary of D.

Side remark: if either of these sequences had a limit point in $\mathbb D$, we would be done already. So this condition does not really add extra information.
Here is a counterexample. Choose an increasing sequence $x_n\nearrow 1$ inductively, so that the hyperbolic distance $\rho(x_n,x_{n+1})$ rapidly tends to infinity. For example, 
$$\log\left|\frac{x_{n+1}-x_n}{1-x_{n+1}x_n}\right|>-\frac{1}{2^n} \tag1$$
Consider the Blaschke products
$$\phi_1(z)=\prod_{k=1}^\infty \frac{z-x_{2k-1}}{1-x_{2k-1}z},\quad
\phi_2(z)=\prod_{k=1}^\infty \frac{z-x_{2k}}{1-x_{2k}z} \tag2$$
Clearly, $\phi_1(x_{2k-1})=0$ and $\phi_2(x_{2k})=0$ for all $k$. On the other hand, there is no  sequence $z_\nu$  converging to $p$, such that  $$\limsup_{\nu\to\infty}\max(|\phi_1(z_\nu)|,|\phi_2(z_\nu)|)<1 \tag3$$
Indeed, from (1) and $\log |\phi_1(z_\nu)|<-\epsilon<0 $ it follows that the hyperbolic distance from $z_\nu$ to the set $\{x_{2k-1}:k\in\mathbb N\}$ remains bounded as $\nu\to\infty $. The same applied to $\phi_2$ implies that the hyperbolic distance from $z_\nu$ to   $\{x_{2k }:k\in\mathbb N\}$ remains bounded. This contradicts our choice of $x_n$.
