Counterexamples to Allen's conjectures

We show that the torus knots $T(2,5)$ and $T(2,9)$ bound smooth Möbius bands in the 4-ball whose double branched covers are negative definite, giving counterexamples to Conjectures 1.6 and 1.8 of Allen in [New York J. Math. 29 (2023) 1038-1059].