Thus far in June, only a trace of precipitation has been reported (on the 2nd and 5th). The last measurable precipitation (.01 inches) fell on May 27th.
Today, however, we have the slightest chance that a rogue, isolated thunderstorm will somehow find a way to put down measurable precipitation at the airport, but it is a long shot. Precipitable water has increased over the past 24 hours and is now nearly 2 cm.
And the dewpoint just hit 41ºF, a welcome change from the obscenely low values of the past couple of days.
Isolated storms generated by this weak surge of monsoon moisture will probably produce gusty winds and perhaps a bit of lightning, but the odds of measurable rain at the airport are low. Looking out further, the latest GFS forecasts suggests we will remain in dry southwesterly flow for the rest of the month. A bit of monsoonal moisture pushes into southeast Utah from time to time (e.g., below), but fails to make it to Salt Lake City.
So, barring a rogue isolated thunderstorm today or a surprise shift in the forecast, we're going to make it through the month without measurable precipitation.