Ultrametric matrices and their inverse

Question

A non-negative square matrix $A$ is ultrametric iff:

1. $A(i,i)>\{A(i,k),A(k,i)\}\forall k,i$
2. $A(i,j)\geq\min\{A(i,k),A(k,j)\}\forall i,j,k$

It is well-known that the inverse of non-negative ultrametric is always diagonally dominant.

Now assume that, we slightly relax the second constraint and change it to:

• If $A(i,j)>0$ then $A(i,j)\geq\min\{A(i,k),A(k,j)\} \forall i,j,k$

In words, the second constraint only holds for non-zero entries. Also for simplicity we can assume that the diagonal of $A$ is all $1$, and other entries are less than $1$.

My question is can we say any of the following about $A^{-1}$:

1. $A^{-1}$ is diagonally dominant.
2. $A^{-1}(i,i) > \sum_{k\neq i, A(i,k)>0} |A^{-1}(i,k)|$

Specially I think the second one should be true.

Any comments will be appreciated.