Lattice and normal-form conjecture for the asymmetric Artin monoid $A_n$

Let $A_n$ be the monoid generated by the images $p_a$ of generators $x_a$, and let $\Delta_n$ be its Garside element. Define left division by $x\leq y$ if and only if $y\in xA_n$. For $a\in\{1,\ldots,n\}$, let $\langle p_2,p_3,\ldots,p_n\rangle$ denote the submonoid generated by $p_2,p_3,\ldots,p_n$.

Lattice and normal-form conjecture.

1. The ordered set $(A_n,\leq)$ is a lattice.
2. Let $x\in A_n$. Then $x\leq\Delta_n$ if and only if there exist $z_a\in\langle p_2,p_3,\ldots,p_n\rangle$ for all $a\in\{1,\ldots,n\}$ such that

$$x=z_n(p_1\cdots p_n)z_{n-1}(p_1\cdots p_{n-1})\cdots z_2(p_1p_2)z_1p_1.$$

These assertions concern the left-divisibility structure and the elements below the Garside element of $A_n$. They are intended to describe a lattice structure and an explicit normal form, but the supplied context gives no resolution evidence.