Is it true or false? (Maybe Feynman would have known, but I don't. If you would have asked me to guess, I would have guessed 6/11.)

I don't know if you're interested in consistency results, but this is certainly true in the Cohen model. Your independent family can consist of $\mathfrak{c}$ mutually generic Cohen reals, and your bijection can map each real in $\Omega$ to something Cohen-generic with respect to that real (which is all but countably many of the Cohen reals in your independent family). The same thing works with random reals.

Maybe Hindman's theorem? mathoverflow.net/questions/360924/…

I suppose the answer is no in $\mathbb R^3$: Start with a disc, and at two distinct points in the interior, cut out a small hole and glue on a long tube (one pointing toward the "heads" side of the disc, one toward the "tails" side), then wrap the tubes around the edge of the disc until you can grow them together in the style of the Alexander horned sphere. (Sorry if this makes no sense at all.) You asked about $\mathbb R^4$, so this doesn't answer your question. But have you looked at generalizations of the horned sphere to dimension $4$?

I don't know if this is the original source or not (or whether you're still interested after 5 years . . . ), but this appears as part (4) of Corollary 3.21 in this paper of Kojman, Milovich, and Spadaro: dkmj.org/academic/nt.product.pdf.

For the record, I would still be interested to know if there is a more hands-on way to do this. For example, is there a Borel partition of the sphere into orthonormal bases? (I can't seem to think of one.)