We describe a search for submillimeter emission in the vicinity of one of the most distant, luminous galaxies known, HerMES FLS3, at z = 6.34, exploiting it as a signpost to a potentially biased region of the early universe, as might be expected in hierarchical structure formation models. Imaging to the confusion limit with the innovative, wide-field submillimeter bolometer camera, SCUBA-2, we are sensitive to colder and/or less luminous galaxies in the surroundings of HFLS3. We use the Millennium Simulation to illustrate that HFLS3 may be expected to have companions if it is as massive as claimed, but find no significant evidence from the surface density of SCUBA-2 galaxies in its vicinity, or their colors, that HFLS3 marks an overdensity of dusty, star-forming galaxies. We cannot rule out the presence of dusty neighbors with confidence, but deeper 450 μm imaging has the potential to more tightly constrain the redshifts of nearby galaxies, at least one of which likely lies at z >~ 5. If associations with HFLS3 can be ruled out, this could be taken as evidence that HFLS3 is less biased than a simple extrapolation of the Millennium Simulation may imply. This could suggest either that it represents a rare short-lived, but highly luminous, phase in the evolution of an otherwise typical galaxy, or that this system has suffered amplification due to a foreground gravitational lens and so is not as intrinsically luminous as claimed.