  In this paper we will make use of the Mackaay-Vaz approach to the universal
$\mathfrak{sl}_3$-homology to define a family of cycles (called
$\beta_3$-invariants) which are transverse braid invariants. This family
includes Wu's $\psi_{3}$-invariant. Furthermore, we analyse the vanishing of
the homology classes of the $\beta_3$-invariants and relate it to the vanishing
of Plamenevskaya's and Wu's invariants. Finally, we use the
$\beta_3$-invariants to produce some Bennequin-type inequalities.
