Going down in absolute sense and going down 30%-40% is a huge matter of degree, no? Look, the TCAD values are wildly inexact--you can't use those to presume fair market value, in spite of the Texas Constitution mandate.
I don't like using sales price as a proxy for value (it's subject to supply distortion), but the Texas A&M home price index shows a price decline of 14% off the peak. I think that's most heavily felt in the $750K-$1.5m range because those homes are upgrades for most buyers and the potential buyers are locked in to cheaper homes by terrifically low rates. I don't know what your property's value is but I'd wager if it's between $400K-$500K any value loss is in the single digits.
Either way, complaining about the city of Austin share of your tax bill while not also putting most of your blame on recapture (if you're getting hit for AISD taxes) is fine, I guess, but the injustice of the latter is in proportion far more offensive to me.