Equality conjecture for the Brunn–Minkowski inequality for the Bernoulli constant

Fix $l\in(0,1)$ and let $\Omega_0,\Omega_1$ be bounded convex domains in $\mathbb R^N$. For $t\in(0,1)$ define

$$\Omega_t=(1-t)\Omega_0+t\Omega_1.$$

Let $\Lambda(\Omega)$ denote the Bernoulli constant associated with a domain $\Omega$. The Brunn–Minkowski inequality is

$$\Lambda(\Omega_t)\geq(1-t)\Lambda(\Omega_0)+t\Lambda(\Omega_1).$$

Brunn–Minkowski equality conjecture. Equality holds if and only if $\Omega_0$ and $\Omega_1$ are homothetic. This is the equality case left open for the Brunn–Minkowski inequality for the Bernoulli constant. The source also mentions, separately, open questions concerning comparison with balls of equal mean width and uniqueness at $\lambda=\Lambda(\Omega)$, but these are not part of this conjecture.