The unfair 0-1-polynomial conjecture

Let $X,Y$ be discrete independent random variables with finite support in the natural numbers, and suppose that $Z=X+Y$ is uniform on its support. Equivalently, let $R(t)=P(t)Q(t)$ be a product of monic polynomials with nonnegative real coefficients, where every coefficient of $R$ is either $0$ or $1$. The unfair 0-1-polynomial conjecture. The random variables $X$ and $Y$ are uniform on their respective supports; equivalently, every coefficient of $P$ and $Q$ belongs to $\{0,1\}$. This is the problem of whether unfairly weighted dice can have a uniformly distributed sum, or equivalently whether a 0-1-polynomial can have a factorization into monic polynomials with nonnegative real coefficients that are not both 0-1-polynomials. The paper studies a first infinite family of cases, but the general problem remains open.