= Solution
For the usual <positive roots>, the chamber condition is that the coefficients of $L_1,L_2,L_3$ be weakly decreasing, since their consecutive differences are the simple-coroot pairings. Adding the same constant to all three coefficients changes no weight, because $L_1+L_2+L_3=0$.
The given coefficients are $(-2,3,0)$. Let $s_1$ exchange $L_1,L_2$, and $s_2$ exchange $L_2,L_3$. With the rightmost reflection acting first,
$$
3L_2-2L_1\xmapsto{s_1}3L_1-2L_2\xmapsto{s_2}3L_1-2L_3.
$$
Thus
$$
\boxed{w=s_2s_1=(1\ 3\ 2),\qquad w(3L_2-2L_1)=3L_1-2L_3=3\omega_1+2\omega_2.}
$$
Its coefficients $(3,0,-2)$ are strictly decreasing, so it lies even in the open <Weyl chamber>; its <Dynkin labels> are $(3,2)$.
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