  We obtain new bounds for the Kakeya maximal conjecture in most dimensions
$n<100$, as well as improved bounds for the Kakeya set conjecture when $n=7$ or
$9$. For this we consider Guth and Zahl's strengthened formulation of the
maximal conjecture, concerning families of tubes that satisfy the polynomial
Wolff axioms. Our results give improved estimates for this strengthened
formulation when $n =5$ or $n \geq 7$.
