TIPS are some of the most complex securities I own, but this is my thinking on this:
Because there is no inflation risk with TIPS, I don't think they will sell at a deep discount when inflation hits.
Let's say you buy a 1.5% 30-year TIPS and a 3.5% 30-year nominal treasury bond right now. In a few years, inflation rises to 10%.
Clearly, the 3.5% 30-year nominal treasury bond will start generating huge negative annual real returns with possibly a risk of negative real return even if held to maturity. Therefore the value of such bond will have to be deeply discounted by the market because of the huge inflation risk they carry.
TIPS, on the other hand, carry no inflation risk. They are guaranteed to generate a positive return over their lifetime, no matter how high inflation gets. In other words, TIPS holders have the option to hold TIPS to maturity while enjoying the inflation-adjusted income generated and recoup their inflation-adjusted principal at the end. So why would they run for the exits and sell at a deep discount when inflation hits? Actually, as an individual TIPS holder, there is no way I would sell my TIPS if inflation was at 10% and rising unless someone paid me a considerable premium for them.
Look at what the VG TIPS fund has done in the past. I bought it very cheaply in November 2008 when inflation was the least of people's worries. Low inflation expectation = cheap fund. Since then, inflation expectations have ramped up, and the fund has returned over 20% while inflation remained tame. High inflation expectation = expensive fund. If inflation starts picking up, inflation expectations will remain high for a while and the fund should continue to do well. When it becomes clear that inflation can been brought under control, then inflation expectations will diminish and the fund will become cheaper again. However, it will most likely become cheaper after the inflation has done its damage.