The gravitational-wave background from supermassive binary black holes ( SMBBHs ) has yet to be detected .
This has led to speculations as to whether current pulsar timing array limits are in tension with theoretical predictions .
In this paper , we use electromagnetic observations to constrain the SMBBH background from above and below .
To derive the maximum amplitude of the background , we suppose that equal-mass SMBBH mergers fully account for the local black hole number density .
This yields a maximum characteristic signal amplitude at a period of one year A _ { yr } < 2.4 \times 10 ^ { -15 } , which is comparable to the pulsar timing limits .
To derive the minimum amplitude requires an electromagnetic observation of an SMBBH .
While a number of candidates have been put forward , there are no universally-accepted electromagnetic detections in the nanohertz band .
We show the candidate 3C 66B implies a lower limit , which is inconsistent with limits from pulsar timing , casting doubt on its binary nature .
Alternatively , if the parameters of OJ 287 are known accurately , then A _ { yr } > 6.1 \times 10 ^ { -17 } at 95 % confidence level .
If one of the current candidates can be established as a bona fide SMBBH , it will immediately imply an astrophysically interesting lower limit .