The DSR created strategical demand for DAIs, but the system needs tactical facility flooding markets with DAIs on demand and working authomatically.
I would propose to give any DSR account a possibility (or an obligaion?) to sell DAIs locked in the DSR automatically if the rate goes up. The target asset may be set by the DSR customer. The triggering rate may be also set by the customer withing the limits set by governance.
It should be economically reasonable for someone collecting 4% annual DSR to get it immediately in case of DAI price spike. It makes also the maker system healthier, automatically flattening spikes.
Although a third-party DeFi may implement this logic, it could be also possible to make it obligatory in DSR: you must sell DAI (with reasonable profit) if there is a system need.