You can find clues on its DPD permit page, but that only relates to the soonest things might go theoretically based upon visible progress. Yes it looks like it could be this month, in theory.
Sometimes a developer slows down (sometimes that's the intent all along) or sometimes the details can bog things down, like a surprise code interpretation that costs in the six figures and means added design work with related delay. Any number of things can get in the way. Financing terms and decisions, construction price and terms, market analysis, any player's insurance details, contamination onsite, a late decision to wring another million out of the price....
Go to
http://web1.seattle.gov/DPD/permitst...spx?id=6424553, find the shoring permit, and watch progress in the "review" tab. When all of the technical categories say "approved" or "waived" then the permit might be in less than two weeks if they choose to jump through the remaining hoops. If you click on "land use permit" you'll see that the decision is published but the permit isn't yet issued.