Dissecting a square into congruent polygons

We study the dissection of a square into congruent convex polygons. Yuan \emph{et al.} [Dissecting the square into five congruent parts, Discrete Math. \textbf{339} (2016) 288-298] asked whether, if the number of tiles is a prime number $\geq 3$, it is true that the tile must be a rectangle. We conjecture that the same conclusion still holds even if the number of tiles is an odd number $\geq 3$. Our conjecture has been confirmed for triangles in earlier works. We prove that the conjecture holds if either the tile is a convex $q$-gon with $q\geq 6$ or it is a right-angle trapezoid.Comment: 19 pages, 11 figure