I know that at r=4, the map exhibits chaotic behaviour for almost every trajectory except for very few which lead to fixed periodic behaviour. For this reason, I am inclined to say that the above question is true as there will exist some initial condition which fulfills the criteria of coming close (in an infinite cycle for example).
However I am hesitant with the wording "generic trajectory" which has me thinking it is referring to any trajectory which as I've mentioned isn't true in a small number of cases. Any advice on this answer would be much appreciated.
The logistic system you are describing has invariant measure $$\mu(dx)=\frac{1}{\pi}\frac{1}{\sqrt{x(1-x)}}$$
So, there is a set $B$ with $\mu(B)=0$ such that any trajectory $\{x_n:n\in\mathbb{Z}_+\}$ with $x_0\in (0,1)\setminus B$ visits any neighborhood of any point in $(1,0)$ infinitely often. This follows for example, from the ergodic theorem which in this case states that for $\mu$ almost any initial condition $x_0$ in $(0,1)$,
$$\mu(A)=\lim_{n\rightarrow\infty}\frac1n\sum^{n-1}_{k=0}\mathbb{1}_A(x_k)$$ So if $\mu(A)>0$, then $x_n$ visits $A$ inifnitely often (the exceptional initial conditions, if there are any, have $\mu$ measure $0$.
Since $\mu$ is equivalent to the Lebesgue measure (in terms of measure $0$), the set of exceptional initial values is a set of Lebesgue measure $0$.
A short description is here