I propose the following uniformly distributed random variable decomposition conjecture:

Assume $X,Y$ are independent random variables supported on a finite subset of the integers, and assume $Z=X+Y$ is uniformly distributed on its support: is it necessarily the case that $X$ and $Y$ are themselves uniformly distributed on their support?

It has been originally posted in post Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients? in a form of polynomial factorization. Based on research there, this is an open problem referenced in Krasner and Ranulac (1937), Sur une propri'et'e des polynomes de la division du cercle, C.R. Acad. Sci. Paris 204, 397--399, where the result is proven for exponents from some arithmetic progression.

The counterexample search might be parallelizable and has been already attempted in various ways (proving non-existence of such counter example by ruling out small degree polynomials, brute-force search, automatic reasoning/proving based on coefficiients, Groebner basis application).