According to Wikipedia the twin prime conjecture is a special case of the first Hardy-Littlewood conjecture: $$\pi_2(n)\sim 2C_2\frac{x}{(\ln x)^2} \sim 2C_2\int_2^n\frac{dt}{(\ln t)^2}$$ where $C_2$ is the twin prime constant. The article states "The conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem and would imply the twin prime conjecture, but remains unresolved." (NB: since Wikipedia does not format "1 / ln t" I have not Mathjaxed it to preserve the quote.)
This language is confusing. I thought that the PNT had now been proved, and that one implication of PNT being true was that "1 / ln t describes the density function of the prime distribution." Moreover, the closing words of the quoted section, "but remains unresolved," seem to refer to the "assumption" about the nature of the density function, although a looser reading of the words might be held to say that it is the twin prime conjecture that remains unresolved.
First question: Why does the Wikipedia language say that this must be assumed? Is this a slight mistatement on Wikipedia's part, or am I missing a subtle distinction between the meaning of the PNT being true and the density function being $\frac{1}{\ln t}$?
Second question: I'm not sure how the twin prime conjecture can be "justified" but "not proven" by a set of circumstances. What specifically is lacking that would be necessary to turn a justification into a proof is not identified in the Wikipedia article. Can someone provide a concise statement of what is missing, and what would suffice to bridge the gap?