Mackey-Glass type delay differential equations near the boundary of absolute stability

For equations $ x’(t) = -x(t) + \zeta f(x(t-h)), x \in \R, f’(0)= -1, \zeta > 0,$with$C^3$-nonlinearity$f$ which has negative Schwarzian derivative and satisfies $xf(x)<0$ for $x≠0$, we prove convergence of all solutions to zero when both $\zeta -1>0$ and $h(\zeta -1{)}^{1\mathrm{/}8}$ are less than some constant (independent on $h,\zeta$). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey-Glass type delay differential equations.