Space of operators acting from one banach lattice to another

In this note we construct a pair of Banach lattices $X$ and $Y$, which have the following properties:
a) $X$ is not order isomorphic to an $AL$-space,
b) $Y$ is not order isomorphic to an $AM$-space,
c) for any continuous linear operator $T:X\to Y$ there exists a modulus $|T|:X\to Y$.
This example refutes the conjecture of Cartwright–Lotz, saying that the negation of at least one of the conditions a) or b) is necessary for the validity of c).