This question is motivated by the MathOverflow question List of properties of Twin primes Dirichlet series and assumes the following definition of the twin prime-power counting function $k_2(x)$ where $\nu (n)$ is the number of distinct primes in the factorization of $n$.
(1) $\quad k_2(x)=\sum\limits_{n\le x}\begin{array}{cc}\{&\begin{array}{cc}1&\nu(n)=1\land\nu(n+2)=1\\0&\text{True}\\ \end{array}\\ \end{array}$
The following figure illustrates a discrete plot of the twin prime-power counting function $k_2(x)$ defined in formula (1) above at integer values of $x$ in blue and the standard twin-prime counting function $\pi_2(x)$ in orange.
Figure (1): Illustration of $k_2(x)$ defined in formula (1) (blue) and $\pi_2(x)$ (orange)
Question (1): Does the twin prime-power counting function $k_2(x)$ defined in formula (1) above take an infinite number of steps?
Question (2): What else is known about the twin prime-power counting function $k_2(x)$ (e.g. asymptotic and error bound)?
Question (3): Is there a reference that provides more information on the twin prime-power counting function $k_2(x)$?
The following two figures illustrate discrete plots of $k_2(x)$ in blue, $\pi_2(x)$ in orange. and $2\,\pi_2(x)$ in green. I've validated that $k_2(x)\le 2\,\pi_2(x)$ for $101\le x\le 10,000,000$ and I'm wondering if this relationship continues to hold for all $x\ge 101$.
Figure (2): Illustration of $k_2(x)$ (blue), $\pi_2(x)$ (orange), and $2\,\pi_2(x)$ (green)
Figure (3): Illustration of $k_2(x)$ (blue), $\pi_2(x)$ (orange), and $2\,\pi_2(x)$ (green)
Note in the previous two figures $k_2(x)$ seems to evaluate fairly close to $\pi_2(x)$. The following figure illustrates a discrete plot of $k_2(x)-\pi_2(x)$.
Figure (4): Illustration of $k_2(x)-\pi_2(x)$