It looks like, someone has put the pdf behind bars (.htaccess)
You don't have permission to access /wp-content/uploads/2016/09/WP-Trouble.pdf on this server.
Apache/2.4.18 (Debian) Server at paulromer.net Port 443
Maybe the traffic from HN raised the flags, i guess.
The basic point is that both Europe and the US have "[...] returned to sort-of full employment at a much lower level of real GDP than informed people projected we’d reach before the crisis struck." 
There are certainly economic implications (eg, should fiscal and monetary policy be tight or loose now), and political implications (is the economy doing really well currently, or not).
EDIT to add: And the bit about the "facts" and "revered leaders" probably alludes to this:
Potential GDP is what a country could produce using all its resources (including machines = capital and employees = labour, multiplied with productivity). Why is there a drop? Maybe productivity dropped, maybe capital dropped, maybe employment dropped. The chart wants to imply that employment dropped. However, the chart goes on to imply, the unemployment rate has gone back to pre-crisis levels. Why is that? Because many unemployed people are not counted, and this is convenient for the current administration. That's what I take the chart and OP to be alleging.
I think the grandparent is implying that the 2018 estimate is modified to suit the preferences of revered leaders (not sure who... Trump? Obama? Yellen? Powell?), presumably to portray the economy as doing well (whereas the 2007 estimate would suggest the economy is still far below its potential). I am guessing at the grandparent's intentions, so it would be good to get clarification.
I think it is a silly claim. One reason for the 2007 estimate to differ from the 2018 estimate is political bias. Another is 11 extra years of data.
And plenty of anecdotal signs suggest the 2018 estimate is clearly better than the 2007 estimate. According to the 2007 estimate, the economy is slowly falling further and further behind its potential. Does anyone think the employment situation is currently worse than 2010? If improving the unemployment rate from 10% to 4% just leads to treading water against potential GDP, how low does it need to go to catch up? -10%? (Unemployment is relevant here because one of the underutilized resources in the economy ~2009 was workers: plenty were without work.)
I take revered leaders to be those who drive the economic models being used to calculate what people are supposed to believe… it could be those figure heads you mention, but it also could encompass a larger swath of people.
>Does anyone think the employment situation is currently worse than 2010? If improving the unemployment rate from 10% to 4% just leads to treading water against potential GDP, how low does it need to go to catch up? -10%?
If one ignores all the changes that have happened to see how the unemployment rate ("where 17 million American workers don't just disappear") is even calculated then sure, the economy is better, because the unemployment rate says so…
Without these standards, science breaks down. Biases slip in. Authority slips in. "Fact" loses its position as the ultimate determinant of scientific truth.
In the deep waters of theoretical physics, the "problem" is that the scientific method gets to be a little abstract. A theory is falsifiable, in principle, but not in practice.
Economics is not a science in the strict, Popperian sense. This essay demonstrates why. Without Popperian standards, you get finger pointing and unresolvable claims of intellectual dishonesty.
This doesn't mean that science is perfect, but it is distinct from nonscientific (including social "science" fields). A history book written in 2068 will, based on no new information, give a totally different account of historical cause and effect to the ones we write today. Darwin's or Newton's differ only inasmuch as new information has become available since hey were written.
This isn't a bash on economics (or history, etc.). Most things are not science.
What truly caused the fall of the Roman Empire. 18th & 19th century imperial historians thought it was debauchery, echoing their zeitgeist about carnal desires and moral rot. Marx thought it was historical materialism. 20th century historians focused more on cultural influx & migrations. Later 20th century historians liked more postmodernist "one damned thing after another" explanations.
The model(s) for linking monetary policy to inflation rates are not science. There is no agreed upon method for one model displacing another one. Of course everyone is accusing eachother of bullshitting.