Positivity criterion for the Green's function of the reflected piecewise-constant problem

Let $I=[-T,T]$, and let $H_{m,M}$ be the Green's function of Problem. For $m\geq 0$, $M\geq 0$, and $m+M>0$, let $\lambda_1$ denote the first Dirichlet eigenvalue of

$$z”(t)=-mz(-t)-Mz([t]), \quad t\in I, \quad z(-T)=z(T)=0.$$

Positivity conjecture. The Green's function $H_{m,M}$ is positive if and only if $M\in(-m,\lambda_1)$.

This conjecture extends the positivity criterion proved in the paper for $T\leq 1$ to arbitrary values of $T$. The proposed extension is motivated only by numerical evidence, so its validity for $T>1$ remains open.