Hierarchy theorem for NTIME intersect coNTIME?
(This would've been a comment, but it doesn't render properly when I try that.)

"I don't even see how to separate" $\;\;\;$$\mathbf{U}$$\mathbf{q}$$\mathbf{uasiLIN}$ $\cap \; \mathbf{coUquasiLIN} \;\;\;$ from

$[$the linear-exponent version of QIP[2]$] \;\; \cap \;\; [$the linear-exponent version of coQIP[2]$] \;\;\;\;\;$.