Yes, but that is only a requirement on what is honestly a derived work (changes made to that BSD-licensed code), as opposed to a requirement that all of the source code in the entire project be licensed under the BSD license.
It's a requirement that applies to works derived from BSD-licensed code, as GPL requirements apply to works derived from GPL-licensed code. It's up to the courts to decide what counts as derived works. e.g. If the courts decide that copying APIs is not fair use, then technically programs linked against BSD-licensed libraries must adhere to BSD terms (although they may also impose additional terms, as this isn't disallowed). On the other hand, if the courts decide copying APIs is fair use then the GPL doesn't apply to programs linked against GPL-licensed libraries (even if it would like to).
GPL 3.0 section 5 part C:
You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged.
This viral component is what the Antiviral License doesn't allow.
Okay, I see what you're saying, but AFAIK, this actually means a whole lot less than you think.
Re "You must license the entire work, as a whole, under this License":
If I take a large BSD-licensed work, and a large GPL3-licensed work (3 because I think 2 might actually be incompatible), and combine them with a little glue, then I must license my "new work" under the GPL3, but the license I am offering really only applies to that little bit of glue, and nothing else. Moreover, there's nothing stopping me from dual-licensing that little bit of glue under a BSD license too, in which case authors of further derived works can choose whether to use my little bit of glue under the GPL3 or the BSD.
None of this makes too much practical difference, because in any case, both the BSD requirements and the GPL requirements apply to the new work, since it contains both BSD and GPL code. The difference is essentially cosmetic. It means I must write that my "new work" is licensed under the GPL, regardless of how little that may mean.
This suggests to users that the work may be used without any conditions that aren't listed in the GPL, which I am required to ensure is true, but for a different reason. What actually requires me to ensure that no further conditions apply besides those listed in the GPL is the explicit requirement that I do precisely that--"You may not impose any further restrictions...". The only way I can do this is by checking that conditions of other licenses I use are also conditions of the GPL.
Re "This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged.", (my emphasis)
I think perhaps the point here is supposed to be that, for instance, a Java program could be distributed in a single .jar archive file, or as a bunch of separate .class files, but it won't necessarily make a difference to what is considered a "work" under law. Even if you distribute a program as separate files, if the courts decide that the program together constitutes a work, then license conditions apply to the program as a whole. I'll have to admit I don't find it all that clear though.
In any case, I actually don't think it matters, because this doesn't seem to be a restriction (e.g. "You must..."), but rather a statement that tries (not particularly well) to explain an aspect of copyright law.
Of course, I am still not a lawyer, but for what it's worth, that's my understanding.