Conjecture: Let $A$ be a $4 \times 4$ row-stochastic, primitive matrix. Let $p_{i}$ be four probabilities such that $p_1 + p_2 + p_3 + p_4 = 1$. Let \begin{align} V &= \text{diag}_i\{p_{i + 1} + p_{i + 3} - p_i \} \\ &= \text{diag}\{p_2 + p_4 - p_1, p_1 + p_3 - p_2, p_2 + p_4 - p_3, p_1 + p_3 - p_4\}. \end{align} Then, $$\rho(A + V) > \rho(A) = 1$$ where $\rho(X)$ denotes the spectral radius of matrix $X$.
Some notes:
- $\rho(A) = 1$ follows directly from Perron-Frobenius.
- $\text{Tr}(V) = 1$.
- $\rho(A + V)$ is a convex function of $V$ (see Cohen 1981.)
- If $p_{i + 1} + p_{i + 3} - p_i > 0$ for all $i$ then the conjecture is true since the derivative of the Perron root with respect to any element is positive (see Theorem 2 of Vahrenkamp 1976).
- If $A$ is doubly-stochastic, the conjecture is true, in fact we have the stronger bound $\rho(A + V) > 1.25$. The proof is the same idea as Lemma 1 of Johnson 1994. I can't quite see how it would generalize to the row-stochastic case since the right-Perron eigenvector will be $\mathbf{1}$, but not necessarily the left-Perron eigenvector.
I have some numerical evidence to support the conjecture, but that's about it.