Optimal second-order SSP coefficient conjecture for fully implicit Runge–Kutta methods

Let an $s$-stage fully implicit Runge–Kutta method have order conditions

$$\text{[the order conditions (OC1)–(OC2) from the source]}$$

and let $r>0$ satisfy the absolute-monotonicity conditions

$$(I+rA)^{-1}e\geq0,\qquad rA(I+rA)^{-1}\geq0,\qquad b^T(I+rA)^{-1}\geq0,$$

with $I+rA$ invertible and $1-rb^T(I+rA)^{-1}e\geq0$, where inequalities are entry-wise. Optimal second-order SSP coefficient conjecture. Then $r\leq 2s$. This is the conjectured upper bound for the SSP radius of an $s$-stage fully implicit Runge–Kutta method of order two; the supplied text gives no resolution status.