Travelers did provide a bond in this case, and that bond is still in existence. Now that the court had ruled against the WTBTS, Travelers will soon be collecting $86K in bond premium from the WTBTS.
Travelers isn't stupid. The way this works is, they sign a document, agreeing to be liable for the entire amount of the judgment. They then cover their risk by taking collateral from the party who wants the bond, so in this case, Travelers is likely to have taken WTBTS real estate as collateral already. Unless the WTBTS fails to pay upon the disposition of the appeal in Conti's favor, Travelers will never have to pay anything out. And if they do have to pay something out, they will be selling WTBTS real estate to cover whatever amount they had to pay.
Emails to Travelers would be a waste of time. They will have no effect on this situation.