Tetrational analogue of Catalan's conjecture: is ${}^2 3-{}^3 2=11$ the smallest positive gap between distinct integer tetrations?

Let ${}^b a$ denote the tetration of the integer $a\geq2$ to height $b\geq2$ (e.g., ${}^3 3=3^{3^3}$).

The smallest positive difference between distinct integer tetrations that I have found through direct computation is $ {}^2 3-{}^3 2=27-16=11. $

This leads me to propose the following tetrational analogue of Catalan's conjecture.

Conjecture.
For integers $a,b,c,d\geq2$ such that ${}^b a\neq{}^d c$, $ \left|{}^b a-{}^d c\right|\geq11, $ with equality only for $\{{}^b a,{}^d c\}=\{27,16\}$ (namely ${}^2 3=27$ and ${}^3 2=16$).

Question.
Can this conjecture be proved or disproved?

This question arose while studying primes representable as differences ${}^b a-{}^d c$ (see here).
A finite search over tetrations below a given bound does not by itself certify all small differences since two much larger tetrations could in principle be very close. This is also why a proof of the conjecture cannot follow merely from checking an initial range of tetrations.

More generally, any effective lower bound for $\left|{}^b a-{}^d c\right|$ as the smaller of the two tetrations tends to infinity would be of interest.

I have not found this specific conjecture in the literature, so I would also be interested in any earlier reference to the statement above.