Here's some more intuition to add to this post. It's interesting to consider what is going on with respect to consumption and labor supply behavior as well. Take my cash-in-advance notes as a starting point.
Here's what happens in a conventional sticky-price New Keynesian model at the zero lower bound. Anticipated inflation is essentially fixed (anchored), and the central bank would like to lower the real rate of interest but cannot do so. Thus, the real interest rate is too high, which makes consumption too low. Firms hire fewer workers "because demand is low," and for the workers to be happy with working less, the real wage must be low.
In my model, consumers need liquid assets to purchase goods. Here, liquid assets include all assets - not just money - that have some use in exchange or as collateral. In the model, the economy can be in a state where there is a shortage of liquidity. The low supply of liquid assets makes consumption low. Indeed we could say that "demand" is low, though I'm not a big fan of that language (as it's misleading in general equilibrium). The people living in this world work today, acquire assets, and consume tomorrow by selling the assets. When there is a shortage of liquidity, firms hire fewer workers, and the workers are reconciled to working less because the return they are getting on their assets is low - that's another way of looking at the high liquidity premium. The real rate of return on assets is essentially determining the effective wage the worker faces, so what's going on in the labor market looks exactly the same as in the New Keynesian model - the real wage is low. What's different is that the real interest rate is not too high, it's too low.
Then, suppose the central bank comes along and does quantitative easing (QE). This increases the stock of liquid assets (say, because long-maturity government debt is worse collateral than short-maturity government debt). This relaxes constraints for consumers, and consumption goes up - it's "demand stimulus," if you like. Firms want to hire more workers, but the workers have to be reconciled to working harder, so their real wages need to go up. This means that the real return they receive on their assets must be higher. At the zero lower bound, the only way that can happen is if inflation goes down.
So, if I were inclined to do it, I could sell that as Keynesian story. The inefficiency in this economy arises from a shortage of liquidity. Relieving that liquidity shortage through QE has the effect of increasing spending and wages. But inflation goes down and the real interest rate goes up. And less inflation is associated with more output. No Phillips curve for sure.