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High-Probability Lower Bound for Sums of Conditional Probabilities in IID Sequences on the Unit Sphere

Let $(\Omega, \mathcal F, \mathbb P)$ be a filtered probability space.
Let $\{u\}_{i=1}^\infty$ be a sequence of i.i.d. RV's, $u_i\sim Ber(\frac{1}{2})\,\forall i\in\mathbb N$, where we define $u_i = 0 \,\forall i \leq 0$. Let $\mathcal F_i$ be the natural filtration $\sigma(u_1, u_2, \ldots, u_i)$, with $\mathcal F_0$ the trivial $\sigma$-algebra. 

Let $\mathbb S^{d-1}$ be the unit-sphere on $\mathbb R^d$ and define $u_{i: i-d+1}:= [u_i, u_{i-1},\ldots,u_{i-d+1}]$. 

I'm interested in the following question: 
do there exist some $\alpha, \beta, c, \kappa>0$ such that the event
$$
\left\\{\sum_{i=1}^N\mathbb{P}\left(|u_{i:i-d+1}\\,\cdot x|>\alpha\mid \mathcal F_{i-1}\right)\geq\beta N\right\\}
$$
happens with high probability ($\geq1-c\exp(-\kappa N)$) for all $x\in\mathbb S^{d-1}$ and all $N\geq 2d$?

Now, I think I solved the question for $d=2$, but I'm having a hard time generalizing the proof. Intuitively, I think the answer should be positive but I can't prove it.

*I'll leave my attempt below.*