Do the 240 roots of E8 implicitly define five disjoint copies of F4 (with 48 roots each)?

Question

Do the 240 roots of E8 implicitly define five disjoint copies of F4 (with 48 roots each)?

Please note that this is an alternative way of formulating the question asked here:

Does a 4_21 exist with 4 vertices from each of of 24 1_22's and 6 from each of 24 "octadeca-diminished" 1_22's (all 48 mutually disjoint)?

and the more general version of this question asked here:

Does the algebraic group E8 ever "collate" two sets of copies of the algebraic group E6?

To see the relationship between this question and those two questions, simply recall that F4 can be obtained from E6 by (Coxeter-Dynkin) diagram folding.

Thanks as always for any time anyone can afford to spend considering this question.