Equality of projective dimensions for discrete modules

Let $\mathfrak{A}$ be an order in a finite-dimensional semisimple algebra, and let $M$ be a discrete object of $\mathsf{LCA}_{\mathfrak{A}}$. Projective resolutions of $M$ in $\mathsf{LCA}_{\mathfrak{A}}$ and in $\mathsf{Mod}_{\mathfrak{A}}$ are compared by their lengths. Projective-resolution length conjecture. It is not possible to find a strictly shorter projective resolution of $M$ in $\mathsf{LCA}_{\mathfrak{A}}$ than in $\mathsf{Mod}_{\mathfrak{A}}$. The surrounding results establish an upper bound in the locally compact module category by the projective dimension in the ordinary module category; the conjectural content is that this bound cannot be improved for discrete modules.