A modification of the uniqueness criterion for the solution of the Watson problem for a half-plane

It is proved that a known theorem yielding the solution of the Watson problem for a half-plane in terms of the Ostrovskii function remains valid if the Ostrovskii function $T(r)=\sup\limits_{n\geqslant0}r^n/m_n$ is replaced by the function $\widetilde{T}(r)=\sup\limits_{r\geqslant x>0}r^x/m(x)$, where for $x\in[n, n+1)$ the function $m(x)=m_n$, or by the function $T^*(r)=\sup\limits_{r\geqslant n\geqslant0}r^n/m_n$.