  We prove upper bounds for the error term of the distribution of squarefree
numbers up to $X$ in arithmetic progressions modulo $q$ making progress towards
two well-known conjectures concerning this distribution and improving upon
earlier results by Hooley. We make use of recent estimates for short
exponential sums by Bourgain-Garaev and for exponential sums twisted by the
M\"obius function by Bourgain and Fouvry-Kowalski-Michel.
