Is a compound aromatic if it also has anti-aromatic rings?
The molecule actually optimizes to a planar, delocalized $\pi$ system

Using B3LYP/6-31G(d) optimizations, we see that the molecule optimizes to become planar. (The initial configuration was intentionally bent.)



Moreover, if I plot the HOMO, it's highly delocalized:



Similarly, the LUMO is also delocalized across the entire molecule:



Consequently, I don't see any indication that the "side" benzene rings are localized from the cycloocta-1,5-diyne ring. The bond lengths are in fairly good agreement with the crystal structure:



Here's the crystal structure from J. Am. Chem. Soc., 1975, 97 (3), 658–659. Note there is a profound triple bond character, but the bridging bonds (1.44 Å) are somewhere between a typical aromatic bond length (1.40 Å) and a full single bond (~1.54 Å).