A Fixed-Point Combinator from B and ω

Let $B$ and $\omega$ be combinators with reduction rules

$$Bxyz \longrightarrow x(yz), \qquad \omega x \longrightarrow xx.$$

An applicative $B,\omega$-term is a closed term formed from these two constants using application only. Does there exist such a term $Y$ satisfying

$$Yx =_{\beta\eta} x(Yx)$$

for every term $x$?