I apologize if this is too elementary a question, but I have not been able to make much progress.

Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. The matrix $A$ is the generator matrix for some continuous-time Markov chain $M$. Define the matrix $$ B = \lambda (\lambda I - A)^{-1}. $$ The matrix $B$ is a stochastic matrix that describes the behavior of $M$ over time intervals of random length $t \sim \text{Exp}(\lambda)$.

Conjecture. The diagonal entries of the matrix $B - B^2$ are non-negative.

The conjecture is not true for an arbitrary stochastic matrix $B$, but in numerical simulations it seems to be true for stochastic matrices of the special form above. I was not able to make much progress toward proving the conjecture, so any ideas are appreciated.