There is a well-known homomorphism from Artin's braid group to (the group of invertible elements of the) Iwahori-Hecke algebra of the symmetric group, or more generally from any Artin-Tits group to the corresponding Hecke algebra. Consider the positive lifts of the elements of the Coxeter group in the Artin-Tits group. Then their images in the Hecke algebra yield the so-called standard basis of the Hecke algebra. Elements of the standard basis have a positive expansion in one of Kazhdan and Lusztig's canonical bases, i.e., have coefficients which are Laurent polynomials with nonnegative coefficients. In the case where the Coxeter group is finite, the positive lifts of the elements of the Coxeter group in the Artin-Tits group are the so-called simple elements of the classical Garside structure. An alternative Garside structure, called dual Garside structure, was introduced for spherical type Artin-Tits groups. One can wonder if the images of these elements in the Hecke algebra still have a positive KL expansion or not. This is especially interesting in type A, as simple dual braids yield a basis of the Temperley-Lieb quotient of the Hecke algebra. We will explain how positivity of images of simple dual braids can be obtained in spherical type using a generalization of Kazhdan and Lusztig's inverse positivity, which predicts that certain elements of Artin-Tits groups, which we call ""Mikado braids"", have a positive Kazhdan-Lustig expansion, together with the fact that simple dual braids are Mikado braids. The positivity of the KL expansion of Mikado braids, shown for finite Weyl groups by Dyer and Lehrer, can be generalized to arbitrary Coxeter systems by adapting a result of Elias and Williamson on the perversity of minimal Rouquier complexes of positive simple braids to a ""twisted"" setting as introduced by Dyer, and asks the question of determining which braids have a minimal braid complex which is perverse.