why Beta(1,1) is an improper prior

Question

While I am looking for the sun-rise problem, in part of prior selection they said that ...

in a section of Beta distribution https://en.wikipedia.org/wiki/Beta_distribution#Rule_of_succession

Jaynes[53] questions (for the uniform prior Beta(1,1)) the use of these formulas for the cases s = 0 or s = n because the integrals do not converge (Beta(1,1) is an improper prior for s = 0 or s = n).

in proof of rule of succession which is using Beta(1,1) as its prior https://en.wikipedia.org/wiki/Rule_of_succession

This improper prior is 1/(p(1 − p)) for 0 ≤ p ≤ 1 and 0 otherwise.[5]

Where the p is an unknown parameter for the probability of success.
Though I don't know where 1/(p(1 − p)) comes from.

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Beta(1,1) is Uniform(0,1) which is proper prior. Apparently, it is integrated into one, thus a proper prior.

Question: why prior Beta(1,1) in proof for the rule of succession is improper?

Answer 1

The Wikipedia passage

Jaynes [53] questions (for the uniform prior Beta(1,1)) the use of these formulas for the cases $s = 0$ or $s = n$ because the integrals do not converge (Beta(1,1) is an improper prior for $s = 0$ or $s = n$). In practice, the conditions $0<s<n$ necessary for a mode to exist between both ends for the Bayes prior are usually met, and therefore the Bayes prior (as long as $0 < s < n$) results in a posterior mode located between both ends of the domain.

is clearly a typo as the rest of the section

Haldane's prior probability (Beta(0,0))

The Beta(0,0) distribution was proposed by J.B.S. Haldane,[64] who suggested that the prior probability representing complete uncertainty should be proportional to $p^{−1}(1−p)^{−1}$ (...)

calls the Bayes-Laplace prior a Beta(1,1) distribution, Haldane's prior a Beta(0,0) [improper] distribution, and Jeffrey's middle ground a Beta(1/2,1/2) distribution.

Ànother sentence that seems to be wrong is

According to Jaynes [53] the main problem with the rule of succession is that it is not valid when $s=0$ or $s=n$ (see rule of succession, for an analysis of its validity).

since Jeffreys' prior, which is supported by Jaynes, has no issue when $s=0$ or $s=n$.