I asked this question on Math Exchange, but as I did not receive a successful answer, maybe you could help me.

Karlin-Rubin's theorem states conditions under which we can find a uniformly most powerful test (UMPT) for a statistical hypothesis:

Suppose a family of density or mass functions $\{f(\vec{x}|\theta):\,\theta\in\Theta\}$ and we want to test $$\begin{cases} H_0:\,\theta\leq\theta_0 \\ H_A:\,\theta>\theta_0.\end{cases}$$If the likelihood ratio is monotone on a statistic $T(\vec{x})$ (that is, for every fixed $\theta_1<\theta_2$ in $\Theta$, the ratio $\frac{f(\vec{x}|\theta_2)}{f(\vec{x}|\theta_1)}$ is nondecreasing on $\{\vec{x}:\,f(\vec{x}|\theta_2)>0\text{ or }f(\vec{x}|\theta_1)>0\}$ as a function of $T(\vec{x})$, interpreting $c/0=\infty$ if $c>0$), then the test of critical region $\text{CR}=\{\vec{x}:\,T(\vec{x})\geq k\}$, where $k$ is chosen so that $\alpha=P(\text{CR}|\theta=\theta_0)$, is the UMPT of size $\alpha$.

In all the proofs I have read (for instance, in page 22 here or in "Statistical inference" by Casella-Berger, 2n edition, page 391), it is (more or less) said: "we can find $k_1$ such that, if $T(\vec{x})\geq k$, then $\frac{f(\vec{x}|\theta_2)}{f(\vec{x}|\theta_1)}\geq k_1$, and if $T(\vec{x})<k$, then $\frac{f(\vec{x}|\theta_2)}{f(\vec{x}|\theta_1)}< k_1$". I would understand that statement if the likehood ratio were strictly increasing, but what about the case in which it is constant?

For example, if $X\sim U(0,\theta)$, the likelihood ratio is monotone on $T(\vec{x})=\max_{1\leq i\leq n}x_i$ ($n$ is the length of the sample $\vec{x}$), but not strictly increasing.

My questions are:

Is the assertion between quotation marks true for every density or mass function with (not strictly) monotone likelihood ratio on $T$?

And what about in the case of the uniform distribution?