McKenna's uniqueness conjecture for large solutions

Let $N\ge1$, $\Omega$ be a smoothly bounded domain of $\mathbb R^N$ and $f\in C^1(\mathbb R)$ a function such that the positivity and Keller–Osserman conditions hold. Assume in addition that the function $\tilde f$ defined by

$$f(t)=\tilde f(t)-\lambda_1t,\qquad\text{for all }t\in\mathbb R$$

is increasing, where $\lambda_1=\lambda_1(-\Delta;\Omega)>0$ denotes the principal eigenvalue of the Laplace operator with homogeneous Dirichlet boundary condition. A large solution is a solution of

$$\begin{cases} \Delta u=f(u)&\text{in }\Omega,\\ u=+\infty&\text{on }\partial\Omega. \end{cases}$$

McKenna's uniqueness conjecture. Under these assumptions, there exists a unique large solution. This conjecture concerns uniqueness for boundary blow-up problems under the condition that the nonlinearity is increasing after subtraction of the principal-eigenvalue term. The source attributes it to P. J. McKenna; no resolution is supplied here.