  The twisted elliptic genera of a $K3$ surface associated with the conjugacy
classes of the Mathieu group $M_{24}$ are known to be weak Jacobi forms of
weight $0$. In 2010, Cheng constructed formal infinite products from the
twisted elliptic genera and conjectured that they define Siegel modular forms
of degree two. In this paper we prove that for each conjugacy class of level
$N_g$ the associated product is a meromorphic Borcherds product on the lattice
$U(N_g)\oplus U \oplus A_1$ in a strict sense. We also compute the divisors of
these products and determine for which conjugacy classes the product can be
realized as an additive (generalized Saito--Kurokawa) lift.
