I mentioned earlier that I would run the numbers to see precisely what employment growth rate would allow the unemployment rate to rise to an average of 4.5% by the fourth quarter of this year (i.e., November in the monthly data) on the assumption that the breakeven employment rate were 40k during the remainder of this year.
A 40k breakeven employment rate implies that the labor force rises at 42k a month. In the interest of simplicity, I assume that the breakeven rate applies to the household survey measure of employment growth, but the distinction is worth only 3k a month, so we can skip that whole discussion.
To get the unemployment rate up to 4.5% by November under these conditions would require that the total employment decline at 33k a month. Assuming that the decline of employment growth were proportional in the private and public sectors, this would imply a drop of 27k a month in private employment, which would take the percent rate of change there to -0.2% (ar) by August, where it would remain. If the June employment gain, being reported tomorrow, is positive, then the subsequent decline would have to be a bit steeper, obviously.
Such employment contractions are typically consistent with the onset of recession, as indicated in the chart below. And it is probably wise to recognize that running employment growth so weak would imply a significantly higher than average recession risk. However, it is also fair to point out that this is a unique episode in the sense that the weakness is (within this thought experiment) being driven primarily from the supply rather than demand side (although the Fed is trying to thread the needle), which presumably would make it less a recession signal.
For me, the stronger point here is that we need to scale back our sense of what sort of employment weakness would provoke a response from the Fed – to revive it. It is more intense, i.e., weaker, than our intuition informed by memory might suggest.
Actual to May with simulation to December

Data are actual to May and simulated to December, to hit a 4.5% unemployment rate by November.
Let me conclude with a simple caveat here. This sort of simulation work assumes inevitably that the household survey measure of employment and the establishment survey measure of employment are similar and it looks past month to month squiggles of the participation rate. Scoring whether the labor market got tighter or easier — and by how much — during June will involve a separate set of calculations. The above exercise gets at the issue of trend employment growth. And it is probably going to be quite low.