The Newton polygon conjecture for tree coefficients

Let $T$ be a rooted tree, let $\sympawn_T$ be its coefficient, and let $\square_T$ be the Newton polygon of the numerator of $\sympawn_T$. Write $\#T$ for the number of vertices of $T$, and call the greatest distance from the root the height of $T$; the number of vertices at height $i$ is the number of vertices of height $i$ in $T$. The Newton polygon conjecture. The upper boundary of $\square_T$ is a horizontal line corresponding to the coefficient of degree $\#T$ in $x$. The lower boundary is a horizontal line corresponding to the constant coefficient in $x$. The right boundary is a line of slope $1$. The left boundary is a sequence of segments of slope $1/i$ for every $i$ between $1$ and the height of $T$. The segment of slope $1/i$ has vertical height equal to the number of vertices of height $i$ in $T$.