I am wondering if $BS(1,4)= \langle a,b:bab^{-1}=a^4 \rangle$ can be embedded into the following group generated by $a_0, a_1, a_2$ with relations:

$a_0 a_1 a_0^{-1}=a_1^4$


$a_1 a_2 a_1^{-1}=a_2^4$


$a_2 a_0 a_2^{-1}=a_0^4$


Remark: the group is constructed in analogy to Higman group