I see what you mean, although not sure it works in English. I think as we are measuring conclusions as units, it should be plural in the English sentence. Possibly: - we do not have any conclusions - we do not have a single conclusion - we do not even have a conclusion Although I think the latter two might translate slightly differently to Polish. Just a thought really.
See this entry from the Cambridge Dictionary. There are two usages of "any" that are relevant here: this article refers to them as "strong" and "weak." The weak form is used in questions or in negative statements, and must always be followed a plural noun, or an uncountable noun. The strong form has a somewhat different meaning and is used only in affirmative statements, and the noun can be either singular or plural.
Since the sentence here is a negative ("we do not have...") and "conclusions" are countable, it can only be in the plural --> "We do not have any conclusions."
But the article says: "We use this meaning of any with all types of nouns and usually in affirmative sentences". So this usage is not limitted only to affirmative sentences. Actually, there is an example of a question. Moreover, "żadnego" doesn't refer to "indefinite quantities" but rather to non-existence (no/none).
Strong form any meaning ‘it does not matter which’
We use any to mean ‘it does not matter which or what’, to describe something which is not limited. We use this meaning of any with all types of nouns and usually in affirmative sentences.
To this meaning corresponds rather "jakiś", "jakikolwiek" or "którykolwiek" (and their other forms).