Only three types of limit of distributions truncated to a finite interval in the upper tail?

Question

Suppose random variable $X$ has a continuous probability distribution with an unbounded upper tail; that is, the CDF of $X$ (call it $F$) is absolutely continuous and $F(x)<1$ for all $x\in\mathbb{R}.$

Now consider the conditional CDF of ${X-a\over w}$ given that $a\le X\le a+w$, for $w>0$:$$G(y,a,w):=\mathbb{P}\left({X-a\over w}\le y\ {\LARGE \mid}\ a\le X\le a+w\right) ={F(a+w\,y)-F(a)\over F(a+w)-F(a)}\,1_{0\le y\le 1}+1_{y>1}.$$

WolframCloud computations lead to the following ...

Observation: Apparently, for all well-known parametric families of such distributions (e.g., Normal, Lognormal, Student's t, Cauchy, Maxwell, Gamma, Gumbel, Weibull, etc.), there are only three types of limit of the CDF of ${X-a\over w}\mid a\le X\le a+w$ as $a\to\infty$: $$\lim_{a\to\infty}G(y,a,w)=\lim_{a\to\infty}\mathbb{P}\left({X-a\over w}\le y\ {\LARGE \mid}\ a\le X\le a+w\right) \\[3ex] = \begin{cases} 1_{y>0} & \text{(Type 1: Degenerate at $0$)}\\[2ex] y\,1_{0\le y\le 1}+1_{y>1} & \text{(Type 2: Uniform on $[0,1]$)}\\[2ex] {1- e^{-y\,w/\beta}\over 1-e^{-w/\beta}}1_{0\le y\le 1}+1_{y>1} & \text{(Type 3: Exponential$(\beta/w)$ on $[0,1]$)} \end{cases}$$ where $\beta$ is a parameter depending on the $X$ distribution.

NB: The Type 3 limit depends on both $w$ and the $X$ distribution, but the Type 1 and Type 2 limits are free of w and free of all parameters of the $X$ distribution, which seems quite remarkable.

NB: Letting $Y_a={X-a\over w}$, for all three types (and any other type, if there are any) we have $$\begin{align}y\le 0&\implies \{Y_a\le y\}\cap\{0\le Y_a\le 1\}=\{Y_a=0\}\implies \lim_{a\to\infty}G(y,a,w)=0\\ y\ge 1&\implies \{Y_a\le y\}\cap\{0\le Y_a\le 1\}=\{0\le Y_a\le 1\}\implies \lim_{a\to\infty}G(y,a,w)=1\end{align}$$ since $\mathbb{P}(Y_a=0)=0$ due to the distribution of $X$ (hence $Y_a$) being continuous; thus, the types of limit differ only for $y$ in the open interval $(0,1)$.

Here are pictures of these three types of limit:

Questions:

1. How generally does this observation hold, and how can it be proved analytically? (Is this something well-known?)

2. What $X$ distributions (if any) do in fact yield a limit not among these three types?

3. In the Degenerate case, the limit is not right-continuous at $y=0$, so it fails to be a CDF. Nevertheless, as $a\to\infty$ the probability mass clearly concentrates in an arbitrarily small neighborhood of $y=0$ (as indicated by the plots below); so, is there a valid interpretation of this as a limiting distribution?

(My original motivating question concerned only the asymptotic variance for just Normal vs. Lognormal. These variances of course follow from the above limiting distributions.)

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Examples

Degenerate Limit

Here are the CDFs (left) and corresponding PDFs (right) for Normal($0,1$), showing the CDFs, PDFs labeled as $G(y,a,w),\ g(y,a,w)$ for various $a$ with $w=1$: Computations show that the limit of the distribution is degenerate at $0$ for apparently any Normal, Gumbel, Rayleigh or Maxwell distribution, or any Weibull distribution with shape parameter $> 1.$

Uniform Limit

Here are the CDFs (left) and corresponding PDFs (right) for LogNormal(0,1), showing the CDFs, PDFs labeled as $G(y,a,w),\ g(y,a,w)$ for various $a$ with $w=1$: Computations show that the limit distribution is Uniform on $[0,1]$ for apparently any LogNormal, Student-t, Cauchy, Levi, Pareto, or Inverse Gamma distribution, or any Weibull distribution with shape parameter $< 1.$

Exponential Limit

Here are the CDFs (left) and corresponding PDFs (right) for Gamma($\alpha=2,\beta=3$), showing the CDFs, PDFs labeled as $G(y,a,w),\ g(y,a,w)$ for various $a$ with $w=1$:

Computations show that the limit distribution is Exponential on $[0,1]$ for apparently any Gamma distribution (which includes the Weibull distribution with shape parameter $= 1$), or any Logistic, Laplace, or Extreme Value distribution.

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For reference, here's the Wolfram code I used:

(* Find limit(a->infinity) Pr[(X-a)/w <= y | a<=X<=a+w] for 0<y<1 *)
FindLimit[distr_,asms_]:=Module[{assms=asms~Join~{0<y<1,a>0,w>0}}, 
G[y_,a_,w_]:=Probability[(X-a)/w<=y\[Conditioned]a<=X<=a+w,X\[Distributed]distr];
Glim[y_,w_]:=Simplify[Cancel[Limit[G[y,a,w],a->Infinity]]];
Print["0<y<1: ", Assuming[assms,Glim[y,w]]];] 

(* Examples: *)

(* In[]= *)
FindLimit[ NormalDistribution[\[Mu],\[Sigma]],{\[Mu]\[Element]Reals ,\[Sigma]>0} ] 
(* Out[]= *)
0<y<1: 1

(* In[]= *)
FindLimit[ LogNormalDistribution[\[Mu],\[Sigma]],{\[Mu]\[Element]Reals ,\[Sigma]>0} ] 
(* Out[]= *)
0<y<1: y

(* In[]= *)
FindLimit[ GammaDistribution[\[Alpha],\[Beta]],{\[Alpha]>0,\[Beta]>0} ]
(* Out[]= *)
0<y<1: (E^(w/\[Beta]) (1-E^(-((w y)/\[Beta]))))/(-1+E^(w/\[Beta]))