Let $A = (a_{ij})$ be an $n \times n$ matrix with entries in the nonnegative real numbers $\mathbb{R}_+$. Suppose that, for each $i = 1,\ldots, n$, the sum $b_i := a_{i1} + \cdots + a_{in}$ of the entries in the $i$th row is positive. Let $$\kappa := \frac{1}{b_1 b_2 \cdots b_n} \cdot \det A.$$

Is $A - \kappa \cdot \mathrm{diag}(b_1,\ldots, b_n)$ a conical combination of rank $1$ matrices in $M_n(\mathbb{R}_+)$ (i.e. $n\times n$ matrices of rank $1$ with nonnegative real entries?)

For example, when $n = 2$, writing $A = \left(\matrix{ a & b \\ c & d }\right)$, a direct computation shows that $A - \kappa \cdot \mathrm{diag}(a + b, c+d)$ is equal to the rank $1$ matrix $\left(\matrix{ \frac{(a + b) c}{c + d} & b \\ c & \frac{( c + d) b}{(a + b)} }\right) \in M_2(\mathbb{R}_+)$.