The following two statements are really "obviously false", but are still open:

$EXP^{NP} \subseteq$ depth-2-$TC^0$

$EXP^{NP} \subseteq$ depth-2-$AC^0[6]$

Just as a reminder:

• $EXP^{NP}$ is exponential time plus an oracle for NP. It contains $NEXP$ (nondeterministic exponential time), $EXP$, and $NP$.

• By "depth-2-$TC^0$" I mean the class of polynomial-size, depth-two circuit families where each gate is an arbitrary threshold function -- i.e., if it has $m$ Boolean inputs $x_1, \dots, x_m$, it is defined by reals $a_1, \dots, a_n, \theta$ and has output 1 iff $\sum a_i x_i \geq \theta$.

• By depth-2-$AC^0[6]$ I mean the class of polynomial-size, depth-two circuit families where each gate is a "standard" $MOD_6$ gate: if it has $m$ Boolean inputs $x_1, \dots, x_m$, it has output 1 iff $\sum x_i \neq 0$ mod 6.

Expanding on the second open problem: Given $A \subseteq \mathbb{Z}_6$, define an $A$-$MOD_6$ gate to be one which outputs 1 iff $\sum x_i \in A$ mod 6.

The most embarrassing open problem in circuit complexity may be the following: Show that for all possible subsets $A$, the AND function requires superpolynomial-size depth-2 circuits of $A$-$MOD_6$ gates.

[PS: Thanks to Arkadev Chattopadhyay for explaining some of these $MOD_6$ problems to me.]