Benjamini–Kalai–Schramm noise-stability conjectures for threshold circuits

Let $f$ be a Boolean function, and let $M$ and $D$ denote the size and depth of a threshold circuit representing it. A circuit is monotone threshold when its threshold gates have nonnegative weights. Benjamini–Kalai–Schramm threshold-circuit conjectures. (i) If $f$ is represented by a monotone threshold circuit of size $M$ and depth $D$, then $f$ is stable under $(1/t)$-noise for $t=(\log M)^{100D}$. (ii) If $f$ is monotone and represented by a threshold circuit of size $M$ and depth $D$, then $f$ is stable under $(1/t)$-noise for $t=(\log M)^{100D}$. The source presents both assertions as conjectures motivated by known results and open questions about bounded-depth threshold circuits.