Yes, another lie. The bandanna, despite being such a simple thing, was far too precious for her to allow herself to simply lose and Jack knew that. He also couldn't let her take the risk herself, it was far too dangerous.

Focused on his climb, he didn't look back as Rialynn pleaded with him to return. The bandanna was almost within his reach, just a few more inches....

And then the little strip of cloth lost its hold on the wing. Jack saw what was coming a moment before it happened, eyes widening slightly. Releasing his grip on the rope he ran a few steps along the airship hull and leaped forward, just catching the bandanna with his fingertips....

... And then vanishing beneath the airship's wing.