$\mathbf{AC}^0$ lower bounds for $\mathsf{Gap}\text{-}\mathsf{Max3SAT}$

Various gapped maximization problems are known not to be $$\mathbf{NP}$$-hard under $$\mathbf{AC}^0$$ reductions, e.g., $$\mathsf{Gap}_{1,\epsilon}\text{-}\mathsf{Max3SAT}$$ (see, e.g., Proposition 4 of Allender-Ilango-Vafa'21; the analogous result for $$\mathsf{MaxClique}$$ is attributed to earlier unpublished work by Arora).

The usual argument for showing that, say, $$\mathsf{Max\text{-}3SAT}$$ requires large $$\mathbf{AC}^0$$ circuits is to $$\mathbf{AC}^0$$-reduce from $$\mathsf{Parity}$$, but for these gapped problems, the $$\mathbf{AC}^0$$ reduction from $$\mathsf{Parity}$$ is precisely what fails. My question is, can $$\mathbf{AC}^0$$ lower bounds for these problems (e.g. for $$\mathsf{Gap}_{1,\epsilon}\text{-}\mathsf{Max3SAT}$$, to make it concrete) be recovered in some other way?