pr.probability – Monotonic coupling between "double-sided Gumbel" distributions

I am interested in finding a monotonic coupling between two random variables. Let $ alpha_1> alpha_2 $, $ b <a $. Define the following two densities (non-normalized):
begin {align *}
f_ {12} (x) &: = exp
left { alpha_2 (a-x) + alpha_1 (x-b) -e ^ {a-x} -e ^ {x-b} right }, \
f_ {21} (x) &: = exp
left { alpha_1 (a-x) + alpha_2 (x-b) -e ^ {a-x} -e ^ {x-b} right }.
end {align *}

trivially, $ int _ { mathbb {R}} f {{}} (x) dx = int _ { mathbb {R}} f {{21} (x) dx $.
Moreover, we can check that
for everyone $ y $, we have
begin {equation *}
int_x ^ { infty} f_ {12} (y) dy ge
int_x ^ { infty} f_ {21} (y) dy.
end {equation *}

Then there should be a monotonic coupling
enter $ f_ {12} $ and $ f_ {21} $.
In other words, we should be able to find a
family of probability measures with densities $ p_x (y) ge0 $
such as:
begin {equation *}
int _ {- infty} ^ x p_x (y) dy = 1, qquad
int_y ^ infty f_ {12} (x) p_x (y) dx = f_ {21} (y)
end {equation *}

for everyone $ x $ and $ y $, respectively.

When $ alpha_1- alpha_2 = $ 1, one of these families is given by
begin {equation *}
p ^ {(1)} _ x (y) = exp left {a-y-e ^ {a-y} + e ^ {a-x} right } mathbf {1} _ {y le x}.
end {equation *}

My question is: can we find a generalization $ p_x ^ {( alpha_1- alpha_2)} (y) $ of $ p_x ^ {(1)} $ that works for everyone $ alpha_1> alpha_2 $?

(Of course we can take measurements with atoms for $ p_x (y) $also – there is no need to limit densities.)