One of the comments in Prove that inequality $AM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN$ gives the suggestion for proving the inequality $$ \sqrt{(x-1)^2 + (y+1)^2 + (z+1)^2}\sqrt{(a-1)^2+(b+1)^2+(c+1)^2} $$ $$ + \sqrt{(x+1)^2 + (y-1)^2 + (z+1)^2}\sqrt{(a+1)^2+(b-1)^2+(c+1)^2} $$ $$ + \sqrt{(x+1)^2 + (y+1)^2 + (z-1)^2}\sqrt{(a+1)^2+(b+1)^2+(c-1)^2} $$ $$ \geq \sqrt{(x-1)^2 + (y-1)^2 + (z-1)^2}\sqrt{(a-1)^2+(b-1)^2+(c-1)^2} $$ for any real numbers $a,b,c,x,y,z$.
Question: Can anyone think of a nice algebraic way to do that?
If you try to put the third product of square roots on the right, then square and have two products, then square again etc. the resulting inequality is with roughly 3800 terms (done with Wolfram Mathematica) and has a mixture of positive and negative terms (so it is not even clear why it holds). Crude applications of Cauchy-Schwarz also do not seem to work.
Any help appreciated!