If inflation rises, that means interest rates will eventually have to rise. That means the discount rate on stocks will increase, which means investors will be willing to pay lower multiples than they are willing to pay currently. So to restate the above, if inflation rises, stocks will go down.
For illustration, a 20-year stream of $100 payments has a present value of $1,805 at a 1% discount rate (=PV(0.01,20,-100)). If the discount rate changes from 1% to 2%, that stream of payments is worth 9.4% less. If the discount rate goes to 3% (still historically low) the present value drops another 9%, or a cumulative drop of 17.6%. If one's portfolio consisting of 25X annual spending went down 25% over the course of a few years because the discount rate hit a whopping 4%, that would be a much bigger effect on one's wealth than the inflation itself - i.e. a loss of $250k on a previously $1M portfolio, compared to spending an extra $1,600/year.
Thus, if one hedges their all-stock portfolio by using options, they are protected from the main consequence of inflation, as well as innumerable other scenarios that could cause stocks to go down. This protection can be bought for 2-3% a year in the case of protected puts, or around 0% if a collar is used. I would expect this price to be cheaper in the long term than a hyper-conservative AA.
If one puts a big chunk of wealth into negative-yielding inflation-indexed bonds that actually lag behind even today's low inflation, then maybe that particular portion of their portfolio avoids the damage from inflation, but it will never compensate for the losses in the other components of one's AA. If the AA is set to 100% inflation-indexed bonds, then one merely loses money to tax inefficiency and the gap between inflation and portfolio performance that
@maizefolk pointed out. Even worse, the sustainable withdraw rate of such a portfolio would be in the 1-2% range (cannot find research on the SWR for hedged portfolios, but it is probably better due to mitigating sequence of returns risk).
TL;DR: Why not use a hedged stock portfolio to protect against the risk of rising inflation, plus
any other risk? The expected cost of hedging is similar to or less than the expected losses of inflation-indexed bonds, which only protect against inflation risk.