$e$-positivity of vertical strip LLT polynomials

In this article we prove the $e$-positivity of $G_{\mathbf{\nu}}[X;q+1]$ when $G_{\mathbf{\nu}}[X;q]$ is a vertical strip LLT polynomial. This property has been conjectured by Alexandersson and Panova, and by Garsia, Haglund, Qiu and Romero, and it implies several $e$-positivities conjectured by them and also by Bergeron. We make use of a result of Carlsson and Mellit that shows that a vertical strip LLT polynomial can be obtained by applying certain compositions of operators of the Dyck path algebra to the constant $1$. Our proof gives in fact an algorithm to expand these symmetric functions in the elementary basis, and it shows, as a byproduct, that these compositions of operators are actually multiplication operators.