In Shakespeare's The Taming of the Shrew, what oath do Tranio (diguised as Lucentio), and Hortensio take together in Act IV, scene ii?

In Shakespeare's The Taming of the
Shrew, Tranio is masquerading as Lucentio, a suitor for the hand of Bianca.
Tranio is actually Lucentio's servant, but Lucentio is pretending to be Cambio, a
teacher for the women in Baptista's housea "job" he has taken to secretly woo Bianca. It
appears that "Cambio" is making headway with Bianca, where no one else has been
successful.

Hortensio, pretending to be a music teacher,
has has enough of this charade. He first admits that he is not a
teacher but a gentleman, and is seems obvious to him that Bianca
prefers Cambio over him, which is disgusting to him as he is a
gentleman and Cambio is a man of lower social standing than Hortensio: a
commoner.

Tranio (in league with Cambio/Lucentio)
commiserates with Hortensio, agreeing the he, too, is disgusted by this turn of events.
He promises that if Hortensio foreswears any further attempt to pursue Bianca, that
Tranio (pretending to be Lucentio) will take the same oath, swearing off anything more
to do with Bianca, who has (at least to Hortensio) been a major disappointment.
Hortensio says:

readability="18">

See, how they kiss and court! Signior
Lucentio,

Here is my hand, and here I firmly
vow

Never to woo her no more, but do forswear
her,

As one unworthy all the former favours
(30)

That I have fondly flatter'd her
withal.

Lorcentio promises to
marry a wealthy woman he has only know for a short while, and insisting that he will no
longer give his love to a woman based upon her beauty, but based
upon the kindness within her. Hortensio further
explains:

readability="24">

Would all the world but he had quite forsworn!
(35)

For me, that I may surely keep mine
oath,

I will be married to a wealthy
widow,

Ere three days pass, which hath as long loved
me

As I have loved this proud disdainful
haggard.

And so farewell, Signior Lucentio.
(40)

Kindness in women, not their beauteous
looks,

Shall win my
love.

The oath the men take
means nothing to Tarino: he is working with Lucentio, however Hortensio's oath further
clears the way for Lucentio (pretending to be (Cambio) to pursue and win the favor of
Bianca.