Lower bound on approximation degree in Nisan-Szegedy
If you believe Lemma 3.8 for $\deg f$, you should also believe it for $\widetilde{\deg} f$. An earlier lemma states that if $f$ is a Boolean function on $n$ variables such that $f(\mathbf{0}) = 0$ and $f(\mathbf{x})=1$ for all $|\mathbf{x}|=1$ then $\deg f \geq \sqrt{n/2}$ and $\widetilde{\deg} f \geq \sqrt{n/6}$. Applying the first part of this lemma,  we conclude the first part of Lemma 3.8, namely $\deg f \geq \sqrt{\operatorname{bs}(f)/2}$. Applying the second part of this lemma, we conclude the second part of Lemma 3.8, namely $\widetilde{\deg} f \geq \sqrt{\operatorname{bs}(f)/6}$.