Stationary-solution and classical-solution conjecture below unit mass

Consider the discrete coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels, with initial data $\rho_0$ satisfying the paper's initial-data condition. Let $m_1(\rho_0)$ denote the initial mass, and let $m$ denote the mass prescribed for a stationary solution. Stationary-solution and classical-solution conjecture.

(i) For $1/2<m<1$, the equation admits a unique stationary solution $\tilde\rho:\mathbb{N}\to[0,\infty)$ such that

$$m_1(\tilde\rho)=m,$$

and $\tilde\rho$ satisfies the stated recursive formula.

(ii) For $1/2\leq m_1(\rho_0)<1$, the equation admits a unique mass-conserving solution $\rho$ for all time, and this solution is classical.

The first part extends the known stationary-solution result from masses at most $1/2$ to the range $1/2<m<1$, while the second asserts global existence and regularity in the same subcritical regime. The source says the extension is motivated by numerical computations; no resolution is supplied.